Would you remove large tonsils/adenoids if 3 yr old mouth breaths, wakes often, moans, chews and sometimes lightly snores in his sleep.

Indication for T &A. Yes, this is certainly a reason to have a t and a.
Maybe. The symptoms you are describing are consistent with signs of airway obstruction. Before removing tonsils and/or adenoids (adenoids are the usual cause), evaluation by an ENT specialist, with a sleep study would be recommended.