In Starcraft 2, after a player drops/leaves on a team, the resources already gathered by that player is split and distributed among allies that remain. But will the resources (mineral/gas) that are continually being gathered be split amongst allies?
Yes, resources gathered will be continuously split among allies.
Also you can control his units after he leaves, train units in his building, and build new buildings in his base using his workers.
Yes. This is sometimes used to perform turbo cheese rushes. Normally, you are not allowed to transfer resources to your allies until several minutes into the game. Dropping out is a way around this. When the ally drops out of a game, all their starting resources plus all minerals gathered by their workers get transferred to the remaining player(s). If I recall properly, a 6 pool can get the initial 6 lings out 25 game seconds faster this way, and have the speed upgrade and plenty of lings pumping faster than normal.
If you are in a team game and see one or more opponents drop out immediately, but the other player doesn't surrender, be prepared to defend an early rush, then immediately counterattack to kill the empty bases of the dropped players. Do not relax and think "oh this will be an easy 2v1" until you've killed off the extra base. The single player using two players' minerals has certain advantages over the remaining team, like the initial head start in minerals, and only having to build tech buildings once. This advantage wanes over time, but can be used to eke out a victory over an unprepared team.