First, yes there are dairy free probiotics - I just Googled "Diary-free probiotic" and came up with lots of hits. I can't recommend one though, since we use Jarro-dophilus.
It is my understanding that if the mom takes a good probiotic the baby will get what it needs through the milk. However, I've also heard it's ok to open a capsule and sprinkle a small amount on your nipple before baby nurses. I'll be interested to see more responses to this.
I gave my DS probiotics from 4 weeks on. When he was that small I just put the powder on my finger and dumped it in his mouth. He liked it. I switched back and forth between New Chapter All Flora, which is the best adult formula out there IMO, and Udos Infant Probiotic which has L. Infantis a baby specific bacteria. It helped a ton with my DS gas and food sensitivities back when he was new. Now at 10 months I give it to him for immune health as well. He has never been sick! People often don't realize that the immune response is initiated in the gut, and the more beneficial bacteria you have the less chance viruses and bacteria have of colonizing you.