I agree that in the case at hand there is a sensible and natural order, but what do you do in a case where there is no such order? Suppose in the dispute the Rabbi mentioned between Tosafot and the Tosafot HaRosh about the obligation to recite HaGomel in the presence of ten and two Torah scholars, which seemingly depends precisely on this question — each one sees a different logical order, and the conclusions contradict each other.

The discussion of whether a certain ruling is a stringency or a leniency is not connected to the order of applying the rules, but rather to the question of which of the two rules you are referring to. One who writes that the ruling not to recite the blessing is a stringency is referring to “do not take [God’s name in vain],” and one who writes that it is a leniency is referring to the laws of blessings.
The halakhic dispute among the medieval authorities (Rishonim) there is not connected to the order of applying the rules. It is a dispute about the rules governing cases of doubt in the laws of blessings (and not about “do not take [God’s name in vain],” or about the order of applying the rules). The question is whether the rule that doubtful blessings are treated leniently applies there as well or not. The implication regarding “do not take [God’s name in vain]” is merely a resulting consequence. And the reason is very simple: regarding the order of applying the rules, there is a clear logic (which you also agreed with). So it is clear that this is accepted by the medieval authorities (Rishonim), or at least there is no reason to assume otherwise..

I’m not sure I understood — in every discussion, must there be only one relevant doubt that one chooses? The order seems logical to me because in one direction it is possible to take both doubts into account (doubtful blessings and then doubt about “do not take [God’s name in vain]”) while the other direction contains only one doubt.
What does a halakhic decisor do if he wants to take both doubts into account? Are there not cases of double doubt, or two doubts, categorically? I didn’t understand by what standard the Rabbi defines “a doubt that applies” as opposed to “an implication that is merely a resulting consequence”?

Another side to the doubt — the Rabbi mentioned, for example, the story about Rabbi Chaim regarding stringency in life-saving situations versus leniency in the laws of the Sabbath. What do you do when it’s stringency against stringency? Say, when two Torah-level laws contradict each other? Seemingly there are all sorts of rules that can be applied here in order to arrive at a “logical” order (for example, passive omission is preferable; what do you do if in one of the laws there was no doubt, and several other possibilities), but is it necessarily true that there can’t be a case where it is left only to the discretion of the halakhic decisor which doubt comes first?

I don’t know what happens in every discussion. Bring a counterexample and we’ll talk.