The carpet drop is just that...a drop...not a jump. You can't jump far enough to hit the water, so don't try. If you just walk off the edge you'll hit a single slime block hidden under the carpet. When you walk off just keep holding forward, if you miss...you'll die.
With the pistons you might want to launch yourself up a couple times to get your timing & aim down. Once you feel you have it, you need to hit the 2nd button when you're at the peak of your launch. You should also hold back after you've been hit just to make sure you make the platform.
The vines aren't that hard, especially since there's a platform to land on if you fail. So you should be able to get multiple tries. You'll want to jump & grab the vines as high as you possibly can. Once you grab them make sure your hand stays to the 1 side of the vines. If you start looking from side to side you'll probably slip off. When you're about to reach the end of the vines on the right, let go of shift & fall to the left & reshift. Again, be careful not to look from side to side. The transfer is tricky at first, but since you can get multiple tries...you should be able to get the hang of it.