Ive shot two deer like that, they got really puffed up with air in the stomach and intestines like you said.

The diarrea like you said dosent sound normal, I have never seen it, but I usually dont cut the intestines very much. I would just wash out the insides especially well with the hose. I used the powerwasher on one and it worked very well to get some crap, fat, and intestines out. (On real low power)

I would not waste the meat unless the liver looked wierd. Thats the best indicator if they are unhealthy to eat.

I would eat it but not eat any rare to be safe. Except mabye the tenderloins. 140 degrees should kill most if not all bacteria a deer would have.