I usually would nurse Kate laying down, but when she's really congested she does better laying next to me with her head on my arm. It's a little awkward to nurse that way, but it definitely helps her breathe while nursing.
Another thing I didn't really know for a long while: if baby gets sick, yeah you'll probably get it too. But if you get sick first and baby doesn't contract it directly, there's a good chance they will avoid getting it because of the antibodies your body is putting into the milk.