Is it advisable to remove the battery while the computer is turned on, and on AC?

I'm not talking about whether or not it is beneficial for the longevity of the battery to have it installed when the computer is running on AC. This is about the act of removal when the computer is running.

I have done that many times and never had problems. The fact that the Power Manager recognizes this after couple of seconds seems to indicate that such a possibility has been foreseen by the engineers.

All user guides for ThinkPads that i have seen, instruct to first turn the notebook off or to hibernate it, then disconnect the AC adapter and all other cables, and finally remove the battery.

you can but it is not advisable especially it is charging. You can remove the battery whilst it is on, by right clicking on the ThinkVantage Power Manager and select 'Remove and Replace battery'.

Well there's option at powermanager to remove battery so it's advisable to do it while its off. But doing it while its on won't hurt so bad