L.Z.
When I first read your post title my first thought was actually, Wow, three months is a little young for the stomach bug! So that answers one of your questions! I'm sure it's not out of the realm of possibility that a three-month old can get this (I have a friend whose newborn had an ear infection and I had never heard of that either, so I guess anything is possible), but in my own experience with my two kids and my friends' kids, they usually wait until about six months to begin with the stomach bugs. You are doing the right thing to keep yourself healthy, and obviously, your baby.
I went back to teaching when both my kids were 3 months old, so I understand where you are coming from. I work with kids all day in a school of about 500 kids, so germs are all around me constantly. Between that and day care, I was always worried about them getting sick (I still do of course) but there is not a whole lot you can do that you're not already doing, and I honestly don't feel that my kids were any sicker than others' kids, even those who did not attend day care. People always told me that they will get exposed to the germs at some point, either in day care or preschool, whichever comes first. It sounds like an excuse and it stinks when they are babies, but that time goes quickly and before you know it, they toughen up, faster than you think. My oldest didn't get an ear infection until she was 15 months old, or the stomach bug until she was almost three years!! My youngest was more susceptible to illnesses more quickly, which frustrated me because I nursed her 100% until she went to day care! I really think that kids are born with a certain constitution for staying healthy. You just never know so you do the best you can. All kids will get sick at some point.
What you can do is keep practicing good hygiene habits for yourself and your baby, but just accept that you can never shield yourself or him from every germ. I totally understand how stressful it is, but I am telling you as someone who has been there, just go with it and do what you can to toughen him up before the germs do. The flu scare right now is definitely stressful - I am dreading the new school year and what it might bring. You can get a regular flu shot for your baby when he is six months old (I know this is a controversial topic for some); both my girls were summer babies so they both had it by the start of hard core flu season their first year of life. Maybe it's being a teacher that gives me this perspective, but when there is something proactive and safe you can do to keep your kids healthy when they are surrounded by other kids, I say, do it.
Good luck, I know it's hard, especially in fall and winter when it seems all you hear about is what virus is going around. Hang in there and keep doing the best you can, it will all work out, I promise!