1. I don't go by genre. I was writing a large review on the good and bad points of each genre, but I don't have time for that now so I'll just leave it at this answer.
A list of my favorite movies to show what I like from each genre:
One Flew over the Cuckoo's Nest (drama)
Fight Club (comedy/thriller)
The Shawshank Redemption (drama)
Once Upon a Time in the West (western)
Lord of the Rings (fantasy)
Le Fabuleux Destin d'Amélie Poulain (comedy/romance)
Se7en (thriller/crime)
Full Metal Jacket (war)
Monty Python and the Holy Grail (comedy)
The Green Mile (fantasy/drama)
Forrest Gump (drama/comedy)
La Vita È Bella (comedy/war)
Howl's Moving Castle (animation/fantasy)
Alien (horror/sci-fi)
Note: This is not a full list. Just one I could put together by following that top 250 list on imdb and picking my favs from it. I could come up with other movies not on that list, but it would make my list too large.
Needs more Tarantino, Kubrick and Coen.
Recently I've been looking a bit more into animated movies too.
2. Yes, I like lotR. Though I don't like how it spawned a large amount of rip-offs and how my father seems to think I'm interested in them because they are like LotR.