Extreme "believers" of any faith don't make sense to me. Religion is not a buffet from which you should pick whatever you like and dismiss what you don't. It's ridiculous to call the people who do that 'moderates' and people who don't cherry pick 'extremists'.
I never said Islam says all gays should be killed. Why do you mention this very specific claim? It is however very clear that homosexuality is considered a sin in both Christianity and Islam. So any Muslim or Christian who says they support gay marriage / homosexuality is a cherry picking hypocrite.
Islams-Clear-Position-on-Homosexuality.pdf (anic.org.au)
Edit: 'And religion changes with time' makes no fucking sense at all. Islam is supposed to be for 'all of humanity'. Why would it change over time? Is how Muhammad acted suddenly not 'good enough' anymore? Are you morally superior to him?
That's the entire problem. Islam is not changing at all. It's only people taking it less seriously. The doctrine itself hasn't changed a bit.
For the extremism part, I believe that the last Prophet in which muslim believe said that : ''Verily, the religion is easy and no one burdens himself in religion but that it overwhelms him. Follow the right course, seek closeness to Allah, give glad tidings, and seek help for worship in the morning and evening and a part of the night'' (that's a sahih Bukhari). It thus seems to me like he recommends to following everything to a T.
As for the changing part, I believe you should inform yourself better of how things work in islamic theology and fiqh. There are different schools that give different weight to different things. There are some who are very litteralist and other are more rationalist. For exemple, a Mu'tazilite like Mohammed Abduh puts a lot more weight on what reason can discover (and believes that the Qu'ran prescribes the use of reason to discover things) and he puts great weight on
maslaha (the common interest), which the Qu'ran, according to him, prescribes that we promote, and so he deduced principles of action by their relation to the good they generated to the people that are concerned with the laws right here and right now (which makes for a dynamic system of laws). There are also jurists who put a lot of emphasis on
ijma' (consensus of jurists), while other put more emphasis on
ijtihad (individual effort at understanding the text). Most don't like
taqlid (blind imitation of precedent rulings) and some (like Jamal Al-Din Al-Aghani) strongly condemn those who do not want to continually renew
ijtihad, (as every society and every epoch has different exigencies, one should always revisit the text and reinterpret it in this new light) and accuse them of
jumud (stagnation). I could also talk a bit about the role of regional customs and the principle of analogy (
qiyas) in interpreting the Qu'ran, but I guess that's enough for today.
I'm not muslim, so I don't know what the current state of affairs is in the high institution of Al-Azhar is. But these little things is what I've gathered from some courses at University. If you want to inform yourself better, I recommend Albert Hourani's ''Arabic Thought in the Liberal Age 1798-1939'' for an overview of how modern Islam came about (it's a little dated, but it's still a classic) and Ahmed Shamsy's ''The Canonization of Islamic Law'' for the basic historical narrative of how Islam is interpreted and what are the basic principle guiding its interpretation.