Do bug bite leave scars?

They may. Depending on the extent of inflammation, your pigmentation, any foreign bodies (such as stingers) andd the degree of envenomation, bug bite will produce varying degrees of scarring from nothing to extensive (black recluse spider).
Not usually. If the bite gets infected it may scar. If it was a venomous bite it may cause tissue injury. Keep it covered with an over the counter antibiotic ointment. If it is not resolved in a week or so seek medical attention.