What is it called when you get up off the ground by pushing from the ground with your hands, similar to The Rock and Shawn Michaels when they bend their legs towards them, then push them and and they land?

Old age.

Or let's go with sarcopenia.

"Sarcopenia (from the Greek meaning "poverty of flesh") is the degenerative loss of skeletal muscle mass and strength associated with aging."
https://en.wikipedia.org/wiki/Sarcopenia

This answer assumes the subject's legs aren't strong enough to rise from the ground without using the hands. Most older people avoid getting down on the ground in the first place.

Now, for the young, strong, and agile, here's how to lie down and stand up:
http://www.mayfieldclinic.com/PE-POSTURE.htm

Springing up off the canvas seems to be called a "kip-up."

"Kip-up (鯉魚打挺 Liyudating) is one of several names for an acrobatic move in which a person transitions from a supine position directly to a squatting position by propelling the legs, and consequently the entire body, away from the floor. It is used in activities such as martial arts, acro dance, breakdancing, professional wrestling, and gymnastics. Because of the flashy appearance of this move, it is often shown in fighting and action movies. The kip-up is also variously called a rising handspring, kick-up, Chinese get up, kick-to-stand, nip up, and carp skip-up.

"From a supine position, the subject draws his legs into his chest (either straight or bent), rolls back slightly and optionally places his hands on the floor next to his ears. The person then thrusts his legs upward and outward while pushing off from the floor with hands and shoulders. The momentum of the thrust carries the body into the air feet first and, with sufficient back arching, all the way up into a squatting position. This is commonly seen by WWE superstar Shawn Michaels, Rob Van Dam and actor and former WWE superstar The Rock."

More variations at:
https://en.wikipedia.org/wiki/Kip-up