Fifteen minutes is extremely unusual. If your win rate is very, very high due to your past as a competitive gamer (like if you're 20-0 or something crazy like that), it could be struggling to find a match for you and giving up, but otherwise I'd think that's a bug or server trouble. Summoner's Rift/blind pick is orders of magnitude more popular than the other maps/modes, which could play a role if you're trying to get the others.
For top champions, some common/powerful tops right now include Olaf, Jax, Elise, Jayce, Darius, or Wukong, but there's many viable ones - it can be a pretty diverse role. For support, Lulu, Taric, Sona, Nunu, or Zyra are among the most popular at the moment, with Blitzcrank being a support who's very frequently banned at mid-Elo. I'd say to take whatever champions might be popular with kind of a grain of salt, though - things can change very quickly, and it could be very different by the time you hit 30, even if that only takes you a month or so.
The only champions I'd outright tell you to avoid are Heimerdinger and Karma, which are slated to be reworked - both are considered mediocre, though may have some use in very niche situations.