1. Do you associate reading particular books with the places you read them or events of the time you read them?
It's more a matter of associating places with books. I'll be sitting in a pizza place or coffeeshop and remember reading a particular book there. If it isn't the subway or home or gym where I normally read, I get little moments of deja vu-like feelings and the title of the book will just pop into my head.
2. Do you remember the books you read or do they fade quickly? Or do you remember some better than others? How about remember details like character names, not just overall plot?
They fade more quickly than I'd like, but some stick better than others. Some I can remember character names, like Leo in Lois Bujold's Falling Free. Others bring forth vivid memories of how I felt when reading them, ie Richard Adams' Shardik.
3. Have you ever forgotten you've read/own a book and borrowed/bought it again?
I haven't forgotten that I've read a book, but I have forgotten I've bought one and have bought it again. Usually, that happens when the book is reissued with a new cover, but not always. Which is why I now carry a typed list (up to 8 pages now) of books to buy.