Can you have a bulls-eye rash and no Lyme exposure?

Yes. Another "bulls eye" rash occurs with "ringworm" , which isn't a worm at all;it is a skin fungus known as tenia corporus.Eassily treated with anti-fungal meds.
Sure. Lots of stuff other than lyme disease can cause annular rashes, including fungus infections, something called erythema multiforme, and a group of conditions called the figurate erythemas, among other things. A dermatologist can sort out the problem for you.