I've raised goats but know nothing on bottle feeding. I bought mine at about 3 months, the normal weaning age. Don't fret though, lots of people have and do bottle feed baby goats and have gone through the same stuff you are going through.
Go to this site: backyardherds.com, there are many experienced goat people on there who have answered these same questions many times in the past.
Fiasco farms is another good rescource. This site is a very extensive and helpful site full of article and article of every question you could possibly have about your new goat. Here, this link should bring you to a page on bottle feeding:
http://www.fiascofarm.com/goats/feeding.htm#bottle
To answer the questions I can answer:
Diahreahea and in activity are big red flags and goat kids can go down pretty quick. While it is normal for bottle babies to get it they need to be pulled through as quickly as possible. I remember one lady said to take them off their food and get some electrolytes in them. But beyond that I don't know much. (please get the full story, I really don't know what I'm talking about when it comes to bottle feeding
Just stuff I've heard off hand)
When your little goat gets to the 3 month mark you can begin to wean with your choice of goat feed. You will have to buy this at a feed store.
And don't worry, few people can resist a goats little face, you aren't the first person to fall for them and not a soul will blame you.
Good luck and God bless with the little guy.
I will be looking up some more links for you. Best thing to do is post your question on Backyardherds and do some reading while you wait for you answer. Try posting some pictures if you can, pictures always help. ...try posting in the emergency section, I think this would count as a mini emergency.