Yes. The late season 9-point I shot last year at the end of rut was in a bad way. Gutted, he weighed only 98 lbs, his antlers broke off when I started moving him, and the meat had a bad odor when we thawed some during the summer to cook. Here's an interesting article:

http://www.deermanagement.us/whitetail-deer-rut-testosterone-poisoning-o...
"Bucks weakened by the additive effects of searching, fighting, cold weather and overall decreased body condition are prime candidates for injuries and secondary infections. To make maters worse, a high testosterone level restricts a buck’s immune system. This means whitetail bucks are susceptible to many more illnesses at time when their hormones are telling them to be more aggressive and less fearful."

Yes, it can happen. I don't think it is a common theme, but it does happen. Probably most don't die during the rut, but later that year where winter starts to take a firm hold. It's partly why after the rut hunting a food source is so productive, as the bucks are trying to put on some last minute fat.

yes very often, also due to fights