1. No, she did not. Like, at all. She actually is still personally against homosexuality. If you read that whole paragraph, I cite sources showing she still is against it but just doesn't want the government to legislate her personal morals. Biden uses the same libertarian excuse to be personally against abortion but not vote to restrict it. On some level I respect this reasoning, but I don't want someone who uses this reasoning to represent me at the highest levels of government where they'll be setting the agenda for LGBT rights, choosing who will defend against legal attacks against LGBT rights, and is in support of fundamentalist Christian activist right wing judges chosen by Trump.
2. There's a difference between not tearing down democratically elected governments and replacing them with dictators (as we have a history of) and stopping dictators that are oppressing their people. I'm not in favor of all our constant U.S. military involvement, but that doesn't mean we shouldn't support our values when we see them taking root in other countries, and do what we can to thwart the aims of dictators, particularly expansionist aims, and human-rights violations. Also, I bet Trump or at least his defenders would use the same logic to defend white nationalists supporting him. It's not a good sign when your efforts exicte fascists and white supremacists. It shouldn't necessarily be disqualifying, but they don't support her for her economic positions, that's for sure.
3. It was the most in depth source I could find. There's sources all over the web, take your pick. She has connections to a cult. I'm not okay with that.
1. I checked all the links, but could't find anything that support your claim that she would still be against homosexuality. The first one is 15 years old and outdated, the Rolling Stone quote is also from that time if you read it correctly. Not sure if you linked the right Ozy video as she's not been talking about homosexuality at any point there. Finally, you seem to think it being impossible that she felt Buescher being left out was because of his religion, which would have violated Article VI of the constitution. However, I agree that was misguided.
Now, for some proofs that she has changed her views on the subject:
from 2012, where she apologizes for her past.
https://www.hrc.org/blog/hawaii-congressional-delegation-calls-for-marriage-equality , where she is part of a movement to bring same-sex marriage to Hawaii, saying that Civil union sare not the same as a reall marriage. This coming from the Human Rights Campaign, which are adamantly in support of same-sex marriage, and the fact that Tulsi has apparently a perfect 100 score with them nowadays shows that she genuinely changed her views on the subject.
Considering her religious Father and thus traditional education (even more so as she got home-schooled), I can see how she got raised staunchly against same-sex marriage, but understand that she has wisened up over the years.
2. The problem of deposing a dictator like that is that it creates a power vacuum, which will get exploited by more extremist groups. We could see that in Irak, in Afghanistan and Lybia. This is also the reason why Bush senior left Saddam in office - he was at that point the lesser of the evils. Also, the democracy the US brings always seems to be based on the very outdated US election system, not necessarily helping the case.
3. I think that is based on a misunderstanding how Hinduism works. In contrary to Christianity for instance, there is no standard, orthodox way to it, and everyone lives his faith differently, each with his own philosophy to it. Hence why there are so many different temples. But sites like the one you linked seem to not understand that that is normal and expected to Hinduism and not a splinter sect like it would be in case of a Judaeo-Christian religion.
She also declines any Hindu meeting where Vishwa Hindu Parishad participates due to it's being a very nationalistic Indian group doing pretty much everything that the article you linked claims her to be doing. https://www.news18.com/news/world/not-participating-in-indias-partisan-politics-us-lawmaker-tulsi-gabbard-denies-chairing-world-hindu-congress-1871215.html
The Nintendo eShop rating Thread: http://gamrconnect.vgchartz.com/thread.php?id=237454