When I get bitten, my bites get pretty big (even if I don't scratch) and sometimes hurt if i move a muscle near the bite. And if I do scratch, it swells a lot and won't reduce size for a few days, even when I use anti-itch cream.
The only one who can tell you for sure about an allergy to mosquito venom is an allergist, who could perform a skin test to find out. However, your symptoms definitely sound like an allergic reaction. To reduce itching after a bite, you might try applying an ice pack or cold compress. Zyrtec, an over the counter antihistamine, is also useful for some people who have itchy skin that's related to allergies.