There is no rule in tennis stopping players from switching the racquet between hands to hit a forehand (or backhand) from both sides. It is extremely rare in tennis (especially among the pro ranks) to see someone who is ambidextrous and can hit a forehand with both hands.
What is more common (but still not very common) is to see a player that hits with 2 hands from both sides. Two female professional players come to mind - Monica Seles and Marion Bartoli. Both hit two-handed forehands and two-handed backhands.
It is probably worth mentioning that in tennis, players that hit a two-handed backhand have an almost ambidextrous skill with their backhand side. This is because most two-handed backhand technique is learned and developed by first practicing hitting a forehand with their backhand side, then later adding their dominant hand to the racquet handle for power and stability. I suspect many professional level players that hit a two-handed backhand could maintain a slow-to-medium pace non-competitive rally by hitting only forehand shots (from both sides).
Sorry I can't answer for the other sports, but I would strongly suspect the same technique is allowed, mostly because of how rare it is for someone to be able to hit a forehand shot from both sides - it's not something that you'd see being used (for any kind of unfair advantage) by more than 1% of players participating in those sports.