I REMEMBER YOU.
You were the person with the awesome coloring! o:
1 - Raise your monsters' friendship with you by brushing them (and feeding of course). The higher the friendship, the more likely to get rarer crops.
2 - Nah, it doesn't matter. Anything can grow anywhere~ There might be benefits to growing certain crops on certain islands, but I don't quite remember, and I never noticed the difference anyway.
3 - Furpuffs (blue chipsqueek/squirrel) do. They're the easiest one to get I think. This
thread shows what monsters grow what. Just search what material or monster you're looking for, and voila, you get your answer~
4 - Again, it's by their friendship levels. There's really no way to know for sure, but the higher their friendship with you, the higher "level" crops grow more frequently, and the lower level ones less often~