So my next questions are going to be rather specific to Shia things.
Theological Differences: So initially the divide between the two major sects were who would be the ideal leader for the Muslim community, Abu Bakr or Ali. Although it seemed to start out as a difference in leadership of politics both present and future (caliphs, ayatollahs, etc), over time it seems that there were more differences in religious interpretation which evolved beyond just this. If possible, I'd like to know the circumstances and origins for the more minor differences and how they came to be, if possible.
The argument about them siding with Ali instead of Abu Bakr and thus creating the rift is just one possible theory.
Some relate the problem to the time when Ali himself became the leader , and the Umayyads contested his leadership.
Another relate to how the Shi'ite declared the Qur'an partially false after the death of Mohammad.
The one I myself believe is the second instance. Because its the one incident all historians agrees upon. A battle did occur between Ali's followers and the Umayyads. And hence it all started.
1. Pictures of Prophet Muhammad: To my surprise I heard that Shi'ite Muslims are more lax in regards to portrayal of the Prophet in images. Whereas Sunnis maintain that visual descriptors for any reason are bad, there are Shi'ites who have no problem if the images are respectful. In Tehran there are pictures of Prophet Muhammad, although Wikipedia says that historically Shi'ites were in agreement with the Sunnis on this issue concerning depictions. I'd like to note ahead of time that near the bottom of the page this article does indeed show images of the Prophet, for those who wish to avoid it.
The prevention of any depiction of God and The Prophets (not just Mohammad), is for two reasons:
The first one is the one you've mentioned. Fear of idolatry.
The second reason is to prevent any ' physical ' mockery of God and his prophets. An image , be it a statue , a portrait ... etc , is prone to vandalism. God is feared , but once one dare to make mockery of him , that would weaken the faith of the followers. As to why the Shia do not care about it. Well they think Ali was the fitted prophet. I suppose from that view point came their leniency towards the depiction of Mohammad.
2. Temporary Marriages: In Twelver Shi'a Islam, there is a tradition known as Nika mut'ah. In short it's a verbal or written contract between two Muslims to undergo marriage, except that it ends at a specified date and time. As far as I know Sunni Muslims don't have an equivalent thing.
Well , to this day , there's supporters and deniers of this type of marriage even among Sunnis.
Some compares it to prostitution , which is prohibited by Islamic teachings , and twisting the basics of marriage to fit into their own visions does not make it any more different. A marriage requires witnesses and a man to formalize the marriage contract as mentioned in an earlier post. Pleasure marriage does not require any of these.
Others considers it normal because its already happening even in my country. Its just that no one need to write a formal date to end the marriage within the contract.
It all boils down to semantics. Do this and do that. Everyone knows the rules , and everyone knows the loop holes.
But it is important to say that Marriage isn't to be taken lightly. Its a religious and social commitment. Treating it like a free pass for a quick roll in the hay just doesn't sound right.
3. Tattoos: Another bit of difference is that the art of tattooing is more relaxed in Shi'a Islam than the Sunnah. For once, Wikipedia was not reliable, giving but a single line claiming that it was permissible and that scholars of the sect believe that there are no hadiths forbidding it. A Google search brought up various message boards and a subreddit by general folks, which may not necessarily be an authoritative source.
Sunnis consider tattoos forbidden , because there's been Hadiths that forbids it. However , no Qur'anic verses backs it up. I guess this is why the SHia are lenient about this one as well.
4. Ashoura and self-mutilation: This is not something universally common to Shi'ites (it's banned in Iran and Lebanon but permitted in India and Bangladesh), there's a belief that infliction of self-harm can aid in the washing away of sins. Traditionally it's done on the Day of Ashura according to Wikipedia. The holiday is practiced by Sunni Muslims as well, but the key difference is that the holiday's a happy event for the Sunni, but a time of mourning for Shi'ites. The exact origins for its practice are not really mentioned except by a saying that its a way of remembering Imam al-Husayn's legacy.
Is voluntary self-mutilation a thing in broader Islam beyond some Shi'ites, and beyond just this holiday?
Self-harm is not allowed in any form for any Muslim. There's countless verses and Hadiths about it. And that whole tradition is just bizarre. Some Shi'ites would tell you its not even true. And harming yourself wouldn't wash away sins. It would get you more sins.
The day of Ashoura is just a day where Muslims fast is following of Mohammad's wish. Basically before Ramadan's fasting became part of the Islamic faith , Muslims were obligated to fast in that day.
Now the day of Ashoura relates to the ' Tenth day of the first month in the Islamic Calender.' Ashoura is a different pronunciation to Asher which translates to ' tenth '.
When Mohammad first arrived to Madinah , he noticed the Jews fasting this day. When he asked why , he was told that the Jews fast in honor of that day when god saved Moses and his people. So Mohammad declared that even Muslims have a right to that honor.
Some say that Mohammad also said that a Muslim need to fast a day before or after Ashoura , just to be different from the Jews. But just like some Hadiths , its weak.
Alternate History/What If? I get the feeling that Shi'ites think about this all the time, but has there been any in-depth consideration or thought experiments (on a professional or amateur level) done on how differently things would've turned out if Ali became the leader of the Muslim community as opposed to Abu Bakr?
History is an interesting thing. You learn about the past and you ponder the endless possibilities that could have happened if one thing or another differs.
I myself do not know if anyone tried to write something of this sort.