1. How many cells are in your 48V battery? Recommendations I've seen for lipo are around 3.0V per cell, anything below 2.5V per cell is dangerously close to permanant discharge. Are you sure the LVC is 44.5V on the 48V controller? I would have expected 42V (10.5V/battery) for lead-acid cutoff.
2. Most lithium chemistries self oxidize and degrade at both high and low state-of-charge. Longest life would be if you kept the battery in the 30-70% range. This effect appears to be less prominent for lithium-iron-phosphate chemistries. So for storage, charge it up to about 1/2 way (then keep in the fridge.)
3. Overcharge protection will be specific to your charger and any
BMS built into the battery. The controller is not involved in the charge cycle.