If you are talking about "high voltage" cells ones made to charge up to 4.35V per cell you generally need a charger that "knows" or is configured to charge up to that voltage before cutting off.
In general adding a bigger battery won't help much but if it is higher energy density (same weight but higher Wh or Ah x V ) then it will fly longer. There is some peak point where the extra ah you're adding (and weight) becomes more than the force the voltage+motors+props is providing and the thing just won't take off, before that it will fly sluggish or like a boat and be more likely to get into oscillations it doesn't have the control authority to stop (or isn't tuned to deal with)