What is the name of this Rummy variant?
It's just one of many forms of a class of games called Contract Rummy. (The various rounds' goals are called contracts).

The major variation here seems to be:

The discard (or rather lack of discard): I guess by your description (since one cannot make three sets with only seven cards) that players just draw a card each turn and are not required (or permitted) to discard, but rather can only play melds and lay off on other melds.

See more common versions at, for example, http://www.rummy-games.com/rules/contract-rummy.html