Worker bee, I don't claim to be an expert but I can tell you what I would do if it were mine. You can take my advice and pass it by your vet.
I'll start wit #4. If he was taken at 6 hours, that's too soon in my opinion unless he was already critical. I leave mine 48 hours to be sure he gets colostrum from the mother. 90% of what he needs, he'll get in the first 12 hoiurs but taking him too soon and he will have poor immune response possibly his entire life. He needs that passive immunity from his mother to kick his immune system in gear. Was the goat milk fresh from a new mother that just had kids? If not, I don't trust colostrum replacers. They are better than nothing, and I use them, but nothing works as well as the mother he's from. Vaccinating the mother's when they're at about the beginning of the third term will give more immunity to their offspring. This passive immunity won't last long once you remove them from the mother. Even left on the mother, they will need vaccinations no later than weaning to build their own immune responses.
You were giving penecillin and it's a good drug for some things but I'd use Nuflor on mine. I use Baytril sometimes but Nuflor is a little better in my opinion. If you know it's respiratory, I'd definitely go with Nuflor or Draxxin. When a fawn has fever it won't want to eat. It's fighting a battle and just like a child they don't feel like eating. Banamine is a great fever reducer but I wouldn't give it longer than 3 days. You can use a water bottle with a drink spout and use cool, not cold, water to squirt in his rectum to drop his temperature (warm, not hot, if temp is low). Fighting high or low temperature works much quicker from the inside. If you can control the temp, I'd use Dexamethasone and B-12 along with the antibiotic. Dex and B-12 will get them up quicker and make them hungry. Don't give Dex and Banamine the same day. I'd stay off Banamine if you're using Dex. Antibiotics aren't necessary (unless the fawn is sick like yours) giving Dex once but more than once I'd definitely give antibiotics with it. It will lower immune response if you give multiple doses. Check the rectal temp every feeding to make sure you keep it under control. Once the Nuflor kicks in and the infection is taken care of, temps should drop. I'd give Nuflor once then wait 48 hrs and give one more dose.
The diarrhea isn't unusual when you're giving antibiotics. You're killing the good and bad bacteria. Give some type of probiotics at least during the antibiotic phase and several days afterwards. I sell Shock Effect products and we have a fawn gel that has charcoal in it. The charcoal will help bind some bad bacterial pathogens like E. coli and carry it out with defacation. We also have a very high bacteria count dispersable powder that mixing extremely easily in bottles. You don't have to use ours but you should be doing something similar.
I hope it helps you. We have saved a lot of fawns but we've lost more than we've wanted to. Let me know if it helps this time or in the future.