My guess:
Will women receive more power in society because they are more sought after?
No, they'll receive even less power, because they become scarce goods and thus commodities and thus less empowered as people. The ability of a guy to get his pick of the women or maintain multiple women at the expense of other guys will become a status symbol, and nowhere in that picture does one see respect or empowerment for the women.
Will societies 'losers' unable to find wives become resentful and prone to violence or substance abuse?
Yep, as has always been the case.
Will new types of un-orthodox relationships become more common?
Depends what you mean by unorthodox.
Will marriage/divorce rates increase/decrease?
See point 1. I'd wager divorce rates will decrease, though marriage rates will increase (women should have no problem finding men).
Will men leave for other nearby countries for a better chance at finding a spouse?
Almost certainly, especially if the alternative is staying in a lonely environmental shithole.