1)Do I have to give her alimony/maintenance for desertion?
-No. There are lot of judgement in which courts denied maintenance in which wife deserts the husband.
2)Do I have to giver her alimony/maintenance for cruelty?
-Same as above.
3)What if I am able to prove both desertion and cruelty?
-You are through to decree of divorce.
4)Since I am higher earning and assuming I prove the desertion and cruelty am I likely to get custody of my sons?
-Yes sure you can get the custody of your sons by proving her a cruel mother or by proving that you can take care better than your wife.