I'll try and take this one at a time
I think it's odd that the theory was "destroyed" by an argument using formal logic. It presupposes that a) humans are logical
I don't see what humans being logical has to do with the conclusions logic comes to.
b) they act in a logical fashion
Using logic doesn't assume humans are completely logical. This also doesn't have to do with the conclusions logic comes to.
c) if stealing is bad then murder is bad is not an opinion
Here you're confusing a valid argument from a sound argument. A valid argument is one where the conclusions necessarily follow from the premises, true or not. A sound argument is a valid one where the premises are true. Here the video was using a sound argument to prove that moral language has logical properties. Not that the example he used is necessarily true. All he's showing is that moral language has logical properties.
d) opinions can't be logically consistent (wtf?)
What's really being said here is that non-cognitivists are saying that "Murder is bad" is really "I disapprove of murder." Sure you can make logical arguments such as: I don't do things I disapprove of; i disapprove of murder; therefore I don't murder. But the issue here is that those arguments aren't of any use within the field of ethics. If you were a non-cognitivist, you can't create a valid argument when you interject "I disapprove of murder" in the middle, since it doesn't necessitate any conclusions valuable within the field.
A 'logical' argument can be made for being homophobic: non-procreation is bad; hence, homosexuality is bad
Again, this is the different between a valid and a sound argument. You listed a perfectly valid argument; the premise, however, might not be (and most likely isn't, but that's another discussion) correct.
not everyone who is homophobic uses arguments, because a bunch also just want to follow scripture
Back to your earlier point here: humans being and using logic doesn't mean certain conclusions are right or wrong. Furthermore, I think they do use an argument without consciously admitting it: I believe the scripture is infallible; scripture says gays are bad; therefore gay's shouldn't marry
this can still be an opinion  and logically consistent.
You're right if you interpret "opinion" as "I believe these first principle are correct"
If I can find a cross-cultural counter-example for something one culture deems moral, and both are equally logical in their claim, what does that say about logic?
I hate to sound redundant here, as I'm sure this must be irritating to read but this comes down to a difference between first assumptions. The arguments can be valid, just not sound.
That is, to go meta-meta: why is something being logical good?
Now that's a good point. There's a whole field dedicated to answering this question (i.e. logic) but my simple answer would be this: Since we're in the field of seeking truth, and there's two essential ways of seeking truth (empiricism and reason (science and logic)) then an answer is good inso far as it uses those methods. Did we define the goalpost here (i.e. Seeking truth through those methods)? Yes, but that's the best I can come up with considering I'm severely under qualified to answer this enormous question. I'd have to give it more thought, but it's interesting to think about!
morality is tied to power, and that b) it is perfectly capable of producing a logical system that can be immoral.
If you haven't already, read some works by Nietzsche, you would like him a lot. But back to the point at hand here: it's one thing to say that humans act because of power, and it's another to say that humans should act because of power (look into the fact/value gap). What you're talking about is psychology of morality or why we act. But again, what we do is not necessarily the way we should act. Also, this is assuming that all moral actions (that is, all actions) are based around power. I would argue that all actions are based around happiness or wellbeing and power is just a means to that end.