I checked my books on the Alapin and found some analysis by Rogozenko which doesn't directly address 14.Qe2, but which you may find useful:

1.e4 c5 2.c3 Nf6 3.e5 Nd5 4.Nf3 e6 5.d4 cd4 6.cd4 d6 7.Nc3 (I agree that this is second-rate; Black should have good chances of a slight edge after this) 7...Nc3 8.bc3 Qc7! 9.Bd2 Nd7 10.ed6 (10.Bd3 de5 11.0-0 is an interesting but probably unsound pawn sacrifice, as in Flores-Karpov, Buenos Aires 2000) 10...Bd6 11.Bd3 0-0 12.0-0 b6 13.Re1 Bb7 14.h3 (14.Rb1 Bf4 15.Bf4 Qf4 16.Re3 Rac8 17.Rb4 Bd5 18.Ne5 Rfd8 19.Qe1 g6 = Hamdouchi-Bagaturov, Biel IZ 1993) 14...Bf4 (Black has good chances to get an advantage; the white pawns on c3 and d4 can easily become targets for Black's pieces) 15.Bf4 (15.Ng5 gave Black the advantage in Ni Hua-Yakovich, Beijing 1997) 15...Qf4 16.Ne5 Ne5 17.Re5 Rac8 18.Re3 Rc7 19.Rc1 g6 20.Bf1 Rd8 and Black was slightly better in Peptan-Bojkovic, Mamaia girls Wch 1991. 

After 14.Qe2, it seems to me that 14...Bf4 with similar ideas to the above games should at least equalise and possibly offer some edge for Black. The idea is to exchange the dark-squared bishops to activate Black's pieces and make it easier to attack White's hanging pawns.

There are some good chapters in books on how to play with and against hanging pawn structures if you are still struggling against hanging pawns; alternatively you can just look up the pawn structure on ChessBase for some ideas.