I think I may have flea bites, but not sure what nasty diseases are transmitted by dog or cat fleas in the u.S.?

None (OK maybe one) No nasty diseases. Common bacteria can enter through breaks in the skin and cause infection(impetigo and cellulitis). Avoid scratching as that can introduce the bacteria. Oral antihistamines (Benadryl) and topical corticosteroids (cortisone) are helpful. If you live in rural New Mexico, Arizona, CA or Colorado there are rare cases of plague from infected rodent flea bites(squirrels, prairie dogs).