These go on the Stack at the same time, and you can choose what order they go in - but each effect is identical, so it really doesn't matter
Exiling two target permanents (any permanents) is a nice way to turn a game around, and even if your opponent can keep chumping it, Ulamog will just mill them out in a couple of swings, anyway
I used to put The Immortal Sun in the majority of my EDH decks, as well, as it is such an overall useful card, but I eventually removed it from those decks, because it is expesnvie and takes too long to provide any significant benefits
So, if a creature with double strike and a creature with first strike have enough damage to destroy each other, they will kill each other in the first damage step
What makes its random ability even funnier is that it is an 0/2, meaning it is likely to eventually kill itself