Runner on 1B. Lefty batter hits bullet grounder to 1B. Runner leaves on contact and is hit by ball (with first baseman behind him). Is the batter out or the runner out? Either way it was the third out of the inning.

Second question...does that batter bat again the next inning?

Runner.
No. Even though the batter-runner was not retired, the batter did complete his time at bat (i.e. became a batter-runner).

Follow up
 

If there were less than two outs, what would happen? Dead ball, batter awarded first base? Or is it a judgment call...a play like that easily could have been a double play...

The ball is dead as soon as it hit the runner and the runner is out for interference. If it were a routine fly ball (fair or foul) that the defense could have caught with ordinary effort, the batter would also be out. The batter in this case receives a fielders choice and their time at bat is completed. With less than two outs this still would be the case. The only ways we would get two here is what I stated earlier about a fly ball or on interference with a thrown ball that in the umpires judgement is an obvious attempt to prevent a double play and occurs before the runner is put out. The immediate succeeding runner shall also be called out. The third way we would get two outs is if a runner has already been declared out or after scoring, a runner intentionally interferes with a defensive player's opportunity to make a play on another runner. In this case the runner closest to home at the time of the interference shall also be declared out.

[Edited by Ed Maeder on Jun 25th, 2004 at 02:08 PM]