Coat catches fire

It's not meant to be literal. There is a fire motif that runs throughout the film: Coat catches fire, Malciek ignites the liquor, the gravestone speaks of "burning", the candles in the tomb. Even the title is a reference to fire--- and in the end (SPOILERS>>>) he dies in a pile of ashes/debris.

There's a few very surreal elements such as the white horse and I don't think the fire theme is an exception. It was obviously not an accident since they could've simply reshot the scene, so I see it as standing for the beginning of this series of events which inevitably ends in ashes.

To fully answer your questions, no bullets cannot set fire to clothes. Maybe modern day tracer rounds (that's a huge maybe), but certainly not any typical lead bullets used back then. Lead doesn't ignite.