There is only an extremely unlikely chance of it damaging the CRT, I wouldn't even bother considering it.
As for the TV coming back on when the mains has been off, there are a number of possibilities:
1) Any 'modern' TV is likely to be remote control, and have a standby option. A 'modern' TV should remember (in EEPROM) the last state it was in, if it was in standby it should come back on in standby, if it was 'on' it should come back 'on'.
2) Older remote control sets had an extra momentrary contact on the on/off switch, when you turn the set on with the mains switch it brings the set out of standby. Following a mains interruption these sets will always revert to standby.
3) Older non-remote sets will simply cme back on when the mains is restored, there was no standby option.
4) A number of the modern 'cheap crap' TV's (usually made in Turkey or Eastern Europe) always come on in standby, even when turned on from the mains switch - so you have to bring them out of stanby, either by the remote, or from a button on the front.