Your racquet can cross the net as long as it never touches it. This does not mean you can just start reaching across the net to hit balls before they come to your side. Before you can hit a return, the ball must first cross over the net to your side of the court. (ITF Rule 24)
No, reaching over the net with your racket is not allowed in tennis. You need the ball to have reached your side of the court first. The only exception to this is if the ball lands on your side and either due to the spin on the ball or the wind, the ball bounces back over the net and onto the other side of the court.
This is why you see players running at a diagonal; to avoid the net or to avoid crossing over that imaginary net/extension. This is the only exception allowed. At low levels of tennis, if you stick your racquet over the net while the point is live, you have invaded your opponent's court/playing area.
Ordinarily, you are not allowed to reach over the net to hit the ball in tennis. Your racquet may cross over the net as long as you make contact with the ball on your side of the net, though. You or your racquet cannot touch the net at all or you lose the point. There are rare cases when you are allowed to reach over the net without losing the point, however.
You need to reach over and tap the ball into their court, without touching the net or their side of the court. If you fail to tap the ball with your racquet and it lands back on their side, they win the point. While this is a weird rule that you probably won't see too often, that one time you do see it, you'll know the rule.
As long as you contact the ball on your side of the net, you can swing through, and your racket will be allowed to cross the net. The second allowance is if the ball is blown back over the net by wind or backspin.
(There has never been any rule in table tennis which stopped you from hitting the ball on your opponent's side of the net, but I understand that there used to be a rule in tennis which said your racket could not pass over the net).
The rule states that when a ball fails to clear the net, or bounces in any spot other than the cross-court service box, it’s a fault. The game stops completely after a fault to contemplate the next action. This is the first fault, so the player is given a second chance.