The bec (battery eliminator circuit) is the part of the esc that supplies power to the receiver and servos, so that you supply all the power from the motor battery, and don't need a separate receiver battery.
Check your esc instructions, there will be a limit on how many servos you can run off the BEC. You can also disable the BEC and use a separate receiver battery instead - you might want to try that and see if you eliminate the problems.
I doubt you ruined your LIPO unless you left everything plugged together for a long time. Most esc's have to be set up for the correct type battery and number of cells, or the correct motor cutoff voltage, so you don't overdischarge the battery. You also do not want to leave the battery connected after you finish flying - the esc and receiver will still draw some power and continue to run down the battery.