My daughter started allergy shots 5 weeks ago. For first time, she's having true allergy like symptoms - runny nose, watery eyes... Is it the shots? She is experiencing runny nose (worsened by: exposure to allergen or irritants, dust, allergens)

Can happen. In the build up phase of allergy shots (the immunotherapy), also depends on the make-up of the allergens used in the mixture of the shot, whether were the shots started pre-seasonal (before the pollen season she is allergic to) or co-seasonal, hence she would have increased symptoms. Any of these symptoms should be communicated to the allergist on every shot day for adjustments to be made, goodluck.