Answer

First of all, congratulations, and may you have abundant joy. I hope and wish that you build a home filled with happiness, wisdom, and joy.
Secondly, the monthly cycle is indeed a problematic matter. Let me begin by noting that the Talmud in Yevamot 64b discusses various presumptive patterns established by three occurrences (see at length Kehillot Yaakov, Taharot, sec. 47), and two opinions are brought there regarding whether a presumption is established even when there is no likelihood of a common cause for all the cases (“mazal causes it” or “the spring causes it”). But the halakhic authorities wrote that in practice there must always be at least some suspicion of a cause. See, for example, in Mekor Chayim, Orach Chayim sec. 467, subsec. 5:

[5] “And if they did not split open,” etc. See Magen Avraham, subsection 12, that if three grains of wheat are found, there is a presumption that there are many more, and perhaps they split open. But Magen Avraham rejected this, saying that we do not say this regarding something that comes from an outside factor. See Chak Yaakov [subsection 25], who challenged this from an established burial ground [Nazir 64b], etc. According to this, we should also be concerned, regarding wheat among which three sprouted grains are found, that perhaps there are many more, since a presumption has been established, and inspection is ineffective, since in a dry mixture there is no blending; if so, they should have to inspect all of them. Further, we should say that since there are three that became leavened, there are many more as well, so that eating the grains as they are, even without having split open, would be forbidden.
Therefore, it seems to me that we speak of a three-time presumption only where there is some causative factor, such that it is reasonable that this should be the case and logic compels such a conclusion. But with something that happens by chance, where there is no reasonably plausible causative factor, there is no presumption for random occurrences. A clear proof of this is from Yevamot 64b regarding a woman whose husbands die: according to the opinion that ‘the spring causes it,’ the rule does not apply to a betrothed woman, and similarly there is no corresponding rule for men. Evidently, wherever there is no plausible causative factor, there is no presumption. Another proof is from Niddah 16a, where they dispute whether fixed cycles are Torah-level or rabbinic. But why should there be a dispute? A three-time presumption is Torah-level, as we find that one may go out into the public domain on the Sabbath with an established amulet [Shabbat 61a]. Rather, it must be that the dispute is whether this is a plausible causative factor. And for that reason it is permitted in the case of sprouted grains, because the fact that rain fell on these is no proof that it also fell on the permitted ones, for perhaps these were from the bottom of the heap, as the Taz wrote in subsection 5 in the name of Rashba. And the question from Nazir is also no difficulty, because there it is plausible that it was designated as a burial ground, since six to eight corpses are laid there. On the contrary, from there there is proof for my words, for if you do not say so, then even if there were not six to eight there, we should say that just as it happened that three dead people were buried there, so too many more were buried there. Rather, certainly we do not say a three-time presumption except regarding something for which it is plausible that it would be so. Accordingly here, with grains that fell into a cooked dish by chance, we do not say a three-time presumption, in accordance with the view of Magen Avraham. That is what appears clearly correct to me.”

One must understand that “suspicion” does not mean something known to be the cause, because then we would not be dealing with a presumption at all but with medical information (a mass that falls to the ground does not do so because of a presumption, but because we know that gravity exists). A presumption always applies when there is something suspected of being a cause—that is, it is possible that there is a cause, but it is not definitely known.
Therefore, at least the Sages thought that there was probably some suspicion of a common cause in the monthly cycle. It is possible that they were mistaken because of lack of knowledge (they had scientific and medical errors). But I once saw that some connected it to the moon. If it causes tides, then it also has effects on the earth. Especially since with human beings there is also psychology, and there is definitely room for the idea that people enter different moods on different dates of the month. And a psychophysical influence can cause physical phenomena.
As for halakhic rulings, I’m not expert in stains, but other questions can be asked and I’ll try to answer.
Again, congratulations,