In other words, have Muslims followed the same path as Christians in the past, rooting out heresy, putting heretics to the stake, etc?
Christianity, since its very inception, has spawned a multitude of weird and wonderful heresies. Is the same thing true of Islam? And were Muslims as fanatical about compelling orthodoxy as Christians were?
Absolutely. Check out this site - it’s an orthodox Sunni site that has a series of articles (“Guard Your Faith”) explaining why every other branch of Islam is wrong.
Oh, my yes. Heresy is always an issue as sects and subsects collide in a actively proseltyzing religion. What actually rises to the level of heresy is of course extremely debateable and has varied enormously - whether the concept is identical to that of Christianity is kind of hard to pin down. In broad sense yes, in a narrow perhaps not. Does Mel Gibson’s father define heresy the same way as the Pope :)?
Shi’ism has tended to provide the most fertile grounds for new religions and minor subsects to emerge ( not just different sects like Zaydi and Isma’ili, but new religions like Druze and Baha’i or nebulous straddlers like the Alawi ), but Sunnism has hardly been immune to such trends. Baha’i is actually a good example - in modern Iran they are generally regarded by the PTB as apostate Muslims and as such have been subject to persecution.