1. There is a Rumor card (can't remember the specific name) that has you spawn a monster on space 19. What if the monster has a spawn also? Which card is followed?
I'm not sure I followed you. But in general, if the monster has a "when spawned, move this monster to..." action, then you spawn the monster on space 19, then move it to where ever it goes, leaving space 19 as it was. So in most cases, the end result is no extra monster on 19.
If you're asking about another card spawning something at the same time, then both cards spawn their monsters.
2. There is a Unique Asset called Walking the Wey Lines, I think, that has you place Gates on it after closing them. The active mystery I had out was to place closed Gates on them. The Walking the Wey Lines did not seem to give me a choice where I wanted to put the closed Gate. The card did not have the word 'may' on it like other unique asset task cards. Can I put the Gate on the mystery and ignore the task asset?
Yes. When you have two actions that happen simultaneously, you choose which one happens first. If you choose "place on mystery", that happens, then the "place on task" would happen... except that there's no gate to place on the task any more, so it doesn't.