2yr old allergic pen? Hives on 2nd day of amoxicillian then no hives for next 8 days of treatment then again on 3rd day after amoxcicillian stopped.

Possible viral hives. It would be unexpected for a true allergic reaction to Amoxicillin not to persist if the medication was continued. Many viral infections can cause hives as a consequence of inflammation involving the allergy arm of the immune system. It would be worth having an allergist test specifically for penicillin sensitivity.
No. Urticaria (hives) in children is such a common condition and is associated not only with food or drug allergy but also with various infections. The clinical presentation of index case does not go along with typical drug allergy.
Maybe, maybe not. Most likely an "amoxicillin rash" a non-allergic rash which occurs when Amoxicillin is used while one has a viral infection. Once the viral infection resolves and Amoxicillin is used again no rash occurs (non-allergic). Patients are commonly incorrectly labeled as Amoxicillin allergic because of this phenomenon. If this was the first time drug was ever used then definitely not true allergy.