c3 was a last desperate measure to transform the R loss in an exchange loss. Black can retire to b6, though. White would better use the tempo to play Nd2, then o-o-o, and try to trap the BQ. In this case, it starts a long tactic sequence that leaves black 5 points up in material. 12. c3, Bg6. 13. Nd2, say black cashes in with NxRh1, allowing white to 14. o-o-o. White is still a tempo behind Nb3, uncovering a terrible trap on the BQ. But white can counterattack with 15. a6, threatening the WQ a move before white does. White will go 16. Qb3, hoping in Na5, 17. Bxf7+, Qf6 (Kf8, 18. Bh6+ (forced), leads to a draw by repetition) , 18. BxQ+, KxQ, 19. Qd5, trying to support the B, c6? (greedily, black wants to take the lsb), 20. Qd6+, KxBf7. Here, white is advised to play 21. Na3. Even down in material, black's position is very cramped, and e4 is hanging. White threatens Ng4+ and RxN, b4, winning the black N, AND f4, opening the way for the remaining rook. Black's best is Be3 (pinning the WN and avoiding Ng4+, but allowing RxN; White will play 22.Qxe5, with a very playable position. White is threatening black's Rh8 and Na4, and if (say) Re8, 23. QxNa4. Quite a try!  

Na3 was the correct answer. A part from this, Bf6, to take black castling rights and blocking any possible f5 idea was to be considered.

White should have interposed BxNc6...

After b5+, Kc3 (best), Ne2+ protects the B and the white king has to go on b3, losing the B after c6, or to b4, allowing black to play the cat and mouse with him  

I think the white king has too many escape squares for there to be mate quite yet. Without a queen or a dark-squared bishop, it's hard to control both e5 and c5. But immediate mate isn't required to win this game. Black's a rook, a knight and a pawn up and can win in any number of ways. I'd say the first priority should be to bring the dormant rooks into the game. 0-0 is an option, or maybe Rc8.