Yes. Many allergies to foods can improve or go away with strict avoidance of the food. Carrot usually causes mouth symptoms after they are eaten raw and is related to cross reactivity with birch tree pollen. Birch tree is the primary allergy in this case. This is called oral allergy syndrome.
Maybe. John hopkins recently published a study in which they found that most childhood allergies do not improve until much later in life. There was about 79% of the teenagers that outgrew their childhood allergies but the other 21% did not. So right now you may just have to wait. You can try improving the immune system through supplements (which the allergist donot recommend) but its worth a shot.
Food allergy. Allergy to any food, in this case carrot will cause symptoms of oral itching, lip or tongue swelling, hives, breathing difficulty and/or throat closure etc. Food allergy is diagnosed with clinical history followed by skin or blood test and that information helps establish the diagnosis and severity of the allergy.