Would toasting a croissant make it catch fire

It's not uncommon for them to catch fire in the conveyor belt toasters common in hotel breakfast buffets. This is because they toast faster than bread, and the process is self-accelerating: the dark (hottest) bits absorb more heat, meaning that they then get very hot, which can ignite the fat, of which there is much more than in bread.

It doesn't help that in those toasters, there's no instant way to stop the process - the element takes a while to cool down after power is removed, and you can't easily extract the scorched bread in time. That's why there are often stern warnings about using them to heat croissants, even though it's perfectly possible to do so with care.

At home, with familiar equipment and your undivided attention, it shouldn't be much trouble, though warming gives a nicer result.

Both.

Croissants have a large amount of fat in it (butter) and lot of thin layers, the external layers will flame up a little bit before burning.