Answer

A. That is indeed the accepted definition, as opposed to a majority of stores, which is a contingent majority (accidental; it could have been otherwise). To the best of my understanding, "most people err in calculating the new moon" is not a natural majority, because it could be different. It is a human error, and people can learn. A majority in nature is something physiological and universal (although I would not rule out the possibility that nature changes over the generations, such as perhaps women giving birth at seven months; but as of now there is a natural majority, and it is not an accidental constellation).
Indeed, one can discuss the majority in a religious court, from which, seemingly, we derive a majority that is before us (in the Hullin passage), and Rabbi Shimon Shkop already asked that this is really a majority that is not before us (for, as Sefer HaChinukh explains, we rely on the fact that the truth is with the majority of the judges—that is, in most panels the majority is correct. That is a majority that is not before us, rooted in the nature of the world and not contingent or accidental. In my humble opinion, this is not comparable to "most people err in calculating the new moon"; examine this carefully).
As an explanation of the matter, I once said a novel idea in a lecture, and I am attaching the relevant section from it:
The possibilities for understanding the nature of majority in a religious court
The Mishnah on 29a, which we saw in the previous lesson, teaches that when there is a dispute in a religious court we follow the majority. Rashi there writes that this is learned from the verse, "Incline after the majority." So too Maimonides, positive commandment 175, and in Laws of the Sanhedrin 8:1, and in Sefer HaChinukh, commandment 78, and elsewhere.
However, in the Talmud this derivation from the verse does not appear explicitly, except regarding an inclined court (in our passage, 3b). The question whether the law of an inclined court is connected to the rule of following the majority was discussed in the previous lesson. For example, in the novellae of Rabbeinu Nissim to 3b it is written that from the law of an inclined court we learn the principle that one must follow the majority.
However, at the end of the passage we see that at least according to Rabbi Yoshiyah, the law of following the majority in a religious court is learned from "Incline after the majority," and from capital cases to monetary cases this is learned by an a fortiori inference. If so, this is an explicit source in the Talmud for the law of following the majority in a religious court. One can then ask whether Rabbi Yonatan agrees with this mode of derivation.
As stated, in the Talmud in Hullin 11a they derive from this verse the general law of majority: "Incline after the majority"—from here we learn that we follow the majority. The intention there is following the majority in clarifying doubts, and not necessarily deciding a legal case in a religious court. Still, it may be the same principle, and that itself may be learned from the religious court.
To understand this, we must examine what the purpose is of following the majority in a religious court. The meaning of the verse, "Incline after the majority," is determined by the problem this rule is meant to solve (the problem that would have existed had the Torah not introduced this rule).
At first glance it appears as in Sefer HaChinukh, commandment 78, who wrote:
The commandment to incline after the majority
To incline after the majority, and this is when a dispute arises among the sages regarding one of the laws of the Torah as a whole, and likewise regarding a particular case—that is, a case between Reuven and Shimon, for example—when the dispute is among the judges of their city, some judging to obligate and some to exempt: one must always incline after the majority, as it says [Exodus 23:2], "Incline after the majority." And by way of explanation our Sages, of blessed memory, said [Hullin 11a]: a majority is of Torah origin.
This choice of the majority appears to be because the two disputing groups are equal in knowledge of Torah wisdom; for it cannot be said that a small group of sages should outweigh a large group of ignoramuses, even if they were as numerous as those who left Egypt. But when the wisdom is equal, or nearly so, the Torah informs us that a greater number of opinions will always agree with the truth more than the minority. And whether they agree with the truth or not in the listener’s opinion, the law requires that we not depart from the way of the majority. What I say—that the preference for the majority is always when the two disputing groups are equal in knowledge of truth—is so everywhere except in the Sanhedrin, where we do not scrutinize, when they disagree, which side knows more; rather we always act according to the words of the majority among them. The reason is that they were a body whose number was mandated by the Torah, and it is as though the Torah explicitly commanded: after the majority of these shall you conduct all your affairs. Also, they were all great sages.
Among the roots of this commandment is that we were commanded in this way in order to strengthen the endurance of our religion. For had we been commanded: fulfill the Torah according to however you can grasp its true intent, then every individual in Israel would say, "My reasoning tells me that the truth of such-and-such matter is so," and even if the whole world said the opposite, he would have no permission to act contrary to what he sees as the truth. This would bring destruction, making the Torah into many Torahs, for each person would judge according to the poverty of his own understanding. But now that we have been explicitly commanded to accept the view of the majority of the sages, we all have one Torah, and our existence is greatly secured by it. We must not depart from their view no matter what. Thus, when we fulfill their directives, we complete God’s commandment. And even if at times the sages, heaven forbid, do not hit upon the truth, the sin will be upon them and not upon us. This is the matter our Sages, of blessed memory, referred to in Horayot [2a]: that when a court erred in a ruling and an individual acted on their instruction, they are liable for the offering and not the individual at all, except in specific cases explained there.
The last paragraph explains why there is any need to decide at all (and in truth this is only where uniformity is required. There is no necessity to decide every dispute, especially since that runs against the value of each person’s autonomy). The previous paragraphs reflect a conception that the majority is generally correct, and therefore we follow the majority. We explained in the lesson that in a democratic majority the situation is completely different. There we follow the majority not because it expresses the truth, but because every person has the right to determine his own fate. See my article in the attached collection.
According to Sefer HaChinukh, is the majority in a religious court a majority that is before us or not before us? It is quite clear that it is a majority that is not before us (as Rabbi Shimon Shkop notes in Sha'arei Yosher 1:1), for we are relying on the reasoning that generally, in most panels, the majority is correct. That does not speak about the panel that is here before us.
 
Rabbi Shimon Shkop’s explanation of majority in a religious court as a majority that is before us
In Sha'arei Yosher 1:1 he elaborates at length in explaining the difference between the two types of majority, and brings a dispute among the medieval authorities as to which of them is stronger. We explained in the lesson Maimonides’ view that a majority that is not before us is stronger, according to the David Levy effect (that one hundred percent of outcomes are determined by the majority of voters). Thus, applying a majority that is not before us works across all cases, and we do not take the minority into account. With a majority that is before us, there are medieval authorities who say that in the end one may not eat all of it, because after all there is a minority before us and one cannot ignore it.
Rabbi Shimon Shkop argues that a majority that is before us does not clarify reality at all (it is a scriptural decree from "Incline after the majority"). His reasoning for this is very puzzling, since he bases it on the fact that every store has a majority against it, which is of course a halakhic argument and not a factual one (a mistake in probability).
He further claims that the words of Sefer HaChinukh are problematic, since according to his approach the majority in a religious court is a majority that is not before us. Therefore he explains the majority in a religious court differently. He claims that we follow the majority of sides. Just as each store casts a side onto the piece of meat (whether it is kosher or not), and we follow the majority of sides, so each judge casts a side onto the case (that the law is such-and-such), and we follow the majority of sides. He explains that this is why both of these are cases of a majority that is before us.
 
An alternative suggestion for explaining Sefer HaChinukh
We saw that a majority that is before us is not based on sampling and generalization, since there is no way to test it experimentally. This is unlike a majority that is not before us, which is a scientific generalization on the basis of a representative sample (such as most women give birth at nine months). Now let us think about the majority in a religious court according to the explanation of Sefer HaChinukh. Suppose the majority is generally correct—what is that assumption based on? Is it the result of sampling and generalization? It seems not, if only because there is no way to carry out checks on such a sample. Let us take a collection of rulings decided by majority against a minority and ask how many were correct. We have no way of knowing what is correct in each case, since we have no more information than the judges themselves. So there is no way to create Sefer HaChinukh’s generalization scientifically by generalizing observations from a sample. So what is it based on? It seems to be based on common sense.
But if so, then this resembles a majority that is before us and not a majority that is not before us. It indeed has the character of a majority that is not before us, but the validity of this majority is based on common sense, like the validity of a majority that is before us. Therefore the Talmud in Hullin determines that the majority in a religious court is similar to the majority of stores, and is a majority that is before us. There is no need to arrive at Rabbi Shimon Shkop’s explanations, and as we saw they are not really correct either. There is in reality a majority in favor of the kosher stores. It is not only a halakhic scriptural decree, and even without the law it is likely that we would have acted this way.
Let us note that the flow of the Hullin passage can remain as Rabbi Shimon Shkop suggested. The difference between a majority that is before us and one that is not before us is that they are different, not that one is strong and the other weak. Therefore the Talmud in Hullin assumes that one cannot be learned from the other. However, in the conclusion—at least according to one explanation in Rashi there—even a majority that is not before us is learned from "Incline after the majority," since in the end it is still following the majority.
B. In the Mordechai there are two answers, and as I recall the difference between them is the question whether the place of fixedness is judged according to the judges or according to the reasonings. My understanding is that Maharatz Chayes means, in the second opinion (that we are discussing the reasonings and not the judges), that reasoning is an abstract thing and has no location, and therefore the notion of the fixed place of the reasoning does not apply.
Perhaps the Mordechai’s intention is like what I said in the lesson I cited above.
[In parentheses: I did not understand your wording regarding a "real mixture" versus "seeing each thing separately." A fixed case too is a real mixture; the question is only whether it is mobile or fixed in its place. Would it enter your mind that the law of fixedness does not apply to dry mixed with dry when each item is recognizable? Judges in a religious court are such a mixture, and so too in the case of throwing a stone into a group, and in the case of frogs. Usually fixedness is specifically with dry mixtures and not liquid ones.]
All the best,