Can I change my playing hand anytime in the game? What is the specific rule for hand change in table tennis? Is it allowed to play using one paddle by holding it any of the hands?
In short, Yes! According to ITTF rules there is no mention of losing the point if you decide to switch your playing hand in between a rally.
However in the description of the question, you have suggested using the other hand with another paddle, that is not allowed. A player may use only one paddle during a point. In addition in case you are switching by throwing your paddle to your other hand you will not be awarded the point if the ball hits the paddle while it is in air. Throwing the paddle to win a point is not allowed and it has to be in full contact with your hand for the point to be won.
I've looked through the rule book and there does not appear to be any rule addressing this specifically. Since it is not addressed, it is safe to assume that it is indeed legal to change racquet hand/free hand during a game, and apparently even during a point.
There are specific rules that apply to raquet hand and others to free hand, but there is nothing specific about that being a static thing (unchanging).
Yes, it is legal to hold the racket with two hands. During the serve you need to hold the racket with one hand, because you need to hold the ball with your other hand.