Can baby get sick if breastfeeding and there is a skin infection in mom?

It depends. There are so many skin infections it is hard to make a blanket statement that covers every variation. It would depend on what the infection is from, how bad is it, where exactly on the breast is the infection, is the infection partly treated and no longer contagious because it is partly treated but the mom is not yet fully recovered. Best bet is to get the opinion of a breatfeeding friendly dr.
Absolutely. Particularly if the infection is on the breast, or within the milk ducts.