Yes. There are a couple of things that a "bull's eye" rash could represent, and most of them should be followed by a medical provider. The most common of these is erythema multiforme, which represents a form of an allergic reaction that your child can get to a viral infection, amongst other things. Best to call your doctor and get it checked out.
Yes. Absolutely, this rash may or may not represent an allergic reaction to the bite, but rather an actual skin infection.