What is this kind of stunt called and what is the highest record for it [closed]

The name of the stunt is a Free Fall. As opposed to using a bungee cord or parachute or cable restraint mechanism, a Free Fall stunt uses only a air bag or other type of landing platform to protect the stuntman from certain squishy death. As water does not compress, a pool is not well suited for very high jumps, especially when it's not a perfect dive. Falling into water from X height is like falling into pavement from X/3 height, as demonstrated on Myth Busters.

As mentioned above, the highest movie free fall is in Sharkey's at 220 feet. 35 years later, the next highest was 125 feet in this year's Assassin's Creed movie.

Both of these are dwarfed by non-cinematic stunts. The same stuntman from Sharkey's did a 313 foot drop from a helicopter. And a few months ago, there was a 25,000 foot Free Fall into a custom designed 10,000 square foot net.

http://ftw.usatoday.com/2016/07/skydiver-luke-aikins-free-fall-stride-crazy-no-parchute-omg

What's the highest even done for a film?

The highest jump till today is done in Sharky's Machine, which is 220 feet. As per source,

At 220 feet, the stunt from Atlanta's Hyatt Regency Hotel (doubling for the Westin Peachtree Plaza) still stands as the highest free-fall stunt ever performed from a building for a commercially released film. The stuntman was Dar Robinson. Despite it being a record-setting fall, only the beginning of the stunt, as he goes through the window, was used in the film. A dummy was used for the outside wide shot of the fall beside the skyscraper.

First Question:

Is there a name for this type of stunt? As opposed to a fall from a plane with a parachute for example.

Yes. It is called Free Fall. In this stunt, the stuntmen don't use any rope or bungee cord. But they use an airbag or a kind of landing platform allowing them to fall and land without getting hurt.