Selecting the proper battery for your robot

Lead acid batteries do build
Lead acid batteries do build up pressure, but they have vents. So long as the Battery is:
A: Not upside down for charging (The pressure will push the fluids out of the vents)
B: Charge voltage is around 13.8V

You can safely charge a SLA battery and not have to worry about overcharging or damaging the battery. They are truly dumb-chemistry batteries. I charge them using a 12V power supply all the time.

No, the battery draws what
No, the battery draws what current it needs, any typical wall adapter rated for 12V will work. (DC of course).