Check your hand position. Are your hands/wrist in a "pushing" position? Or are they in a "punching" position? If it's the latter, then that's the problem.
Just imagine yourself doing push-ups, but using dumbbells instead of the ground. You should be pushing the weights up with the middle/bottom portion of your palms, directly over the wrists.
Also, add some forearm and grip exercises to your routine. This will prevent you from "strangling" the weights, which can be another cause for wrist/hand discomfort during pressing exercises.
As for your triceps, it sounds like bad form. Reduce the weight a little. When you bring the weights down, make sure your upper arms and elbows are away from your body. And stop when your upper arms are parallel to the deck. Some guys go all the way down. But that puts unnecessary stress on your shoulders. Everyone is built differently, so this might not be a problem for you.
On the way up, just push the dumbbells straight up, as far as you can. Reach to the top. Don't arc the dumbbells and try to make them touch. Because then you will be doing a dumbbell fly.
Something like this..