How many companions can you have? I had difficulty getting two at the same time in the past, and never tried again. However, right now I have ED-E "and" the sniper following me. What's the limit?

I like the face of my previous character. Is it possible to place it on my new character, some how?

You can have one human and one non-human companions (total of 2)...but you can't have two human or two non-human companions together.

The face - once your chr. is created, no. Unless you're on PC and are very good at modding/using the GECK face creator to recreate exactly what you had before, perhaps.
If you haven't created your chr. yet...then it's just a matter of picking the same "look" options as you did before, as closely as you can remember.