Is it legal in racket sports to pass the racket from one hand to the other?

Regarding squash, it is perfectly alright to switch your hands while playing a shot. You can use both your hands too. Here is a video which shows that it is allowed.

It is perfectly legal in table tennis to change the racket to other hand during a point. Though uncommon and extremely difficult, it does provide a few exciting moments for the spectators.

ITTF has compiled a video capturing these exciting moments.

And if it is legal for a point, it should be legal for a set and a match as well.

In tennis, rule #24, which covers all the scenarios in which a player loses a point, does not forbid a player from switching the hand that holds the racket.

24. PLAYER LOSES POINT

The point is lost if:
a. The player serves two consecutive faults; or
b. The player does not return the ball in play before it bounces twice
consecutively; or
c. The player returns the ball in play so that it hits the ground, or before it bounces, an object, outside the correct court; or
d. The player returns the ball in play so that, before it bounces, it hits a permanent fixture; or
e. The receiver returns the ser vice before it bounces; or
f. The player deliberately carries or catches the ball in play on the racket or deliberately touches it with the racket more than once; or
g. The player or the racket, whether in the player’s hand or not, or anything which the player is wearing or carrying touches the net, net posts/singles sticks, cord or metal cable, strap or band, or the opponent’s court at any time while the ball is in play; or
h. The player hits the ball before it has passed the net; or
i. The ball in play touches the player or anything that the player is wearing or carrying, except the racket; or
j. The ball in play touches the racket when the player is not holding it; or
k. The player deliberately and materially changes the shape of the racket when the ball is in play; or
l. In doubles, both players touch the ball when returning it.

To address your second question:
Rule #24.j. states if the ball in play touches the racket when the player is not holding it, the player loses the point.

USTA Regulations IV.D. Table 17 provides an official with guidelines on when to assess a penalty for racket abuse.

The guidelines state an official should always penalize throwing a racket that, even inadvertently, strikes a person.

The guidelines state an official should generally penalize:

• Throwing a racket into the fence or net
• Throwing a racket in anger or in any manner that could or does endanger any person or that damages any part of the court surface or equipment
• Breaking a racket

The guidelines state an official should sometimes penalize:

• Tossing a racket into the air
• Tossing a racket at the player’s bag
• Bouncing a racket on the court

The complete Friend at Court can be read here.

Yes, it is LEGAL to switch racket "handed-ness" during play. From the USTA web site:

Q. I am a left-handed tennis player. During play, I have found that I can hit the tennis ball almost as well with my right hand as I do with my left. Do USTA rules forbid players to switch hands during play?

A. No. You can play with either hand. In fact, three-time USTA national champion Marty Devlin of Trenton, NJ has employed this “two forehands” style since he started playing tennis. The old joke when people were preparing to play Marty was to hit to his backhand. But, alas, he has no backhand.

Also, I have cousins in the Miami area who have played tennis with Jai alai players who will switch handed-ness frequently even during the same point / rally.

I know that in tennis you are allowed to use either hand to hit the ball with the racquet during any point, set or match. You may not use more than one racquet during any single point, though (though you are allowed to switch racquets between points). And racquet throwing (for the purpose of hitting the ball) is not allowed - the racquet must be in your hand at the point of contact or you lose the point. It is perfectly legal though, between shots, to drop the racquet and pick it back up and continue the point.