Mono has a good bit of stretch. Braid has almost none. Stretch is kind of a built-in "shock absorber". This can be helpful if you're fighting a big fish near the breaking limit of the line. Line with little or no stretch transmits bites better. I generally use mono but use braid for certain situations.
Memory is bad. It means the line wants to retain the shape of the spool, so it will come off in coils when casting. Limp lines are better.
Could be a lot of things: cheap line, cheap reel, reel too small, line too big. It also sounds like you may have too much line on the spool. Can you give us more details?