I think 5...bxc6 was a mistake for him. The ordinary move is ...dxc6, because that leads to Nxe5 Nxe4 Nxe4 Qd4 and Black gets the material back.
"Queen cannot be moved since it will lead to mate in two" - It's actually not mate in 2 since you don't have to recapture, but it IS very bad, so you're better off taking the 2 rooks for the queen, like you did.
Strictly speaking, you were winning when you took the draw (you could have moved your king to h1 instead of f1 to avoid the threat of Qxh2) but I can understand why you took it.
Berlin Defence characterised by:
1. e4 e52. Nf3 Nc63. Bb5 Nf6
has a reputation of a solid & a drawish opening. In recenlt concluded chess championship match at Sochi it was played in Game # 2, 7, 9 & 11.
While this opening is common at GM level- its seldom played at sub 1400 rated games. Luckily, during Anand vs Carlsen match- I got some nuance to play Berlin defence which I put to some good use against an opponent today (25/12/2014).
Footnotes: (1) If at move #4- If white castles 0-0 then 4....Nxe4 5. d4 Nd6 6. Bxc6 dxc6 7. e5 Nf5 8. Qxd8 Kxd8 (taking off the queen's)- leading to very drawish game.
(2) I played at move #4:Nc3 is second most popular move after castling.