ok im clearing things up 

no you cant put monsters in house. no u cant train them .  all u do is feed them fodder and take them to new island but not for awhile so please stop asking

when you have captured a monster you can use it a battle if you train it

Just don't capture any yet.  The monstology toplist should give you a clue - nobody's doing it at all except a few people who have caught monsters and don't know what to do with them but the monsters haven't died yet.

got no clue were u heard that from but ur wrong

I think you take them to your house

8 months!

you take them to the next island but not for like 8 months

nothing yet