Impeachment is only appropriate for "high crimes and misdemeanors." To my knowledge, there is no statute or case law establishing that what the NSA is doing is a crime, much less a "high crime." The only ruling I'm aware of is under appeal, and any vote--even if it weren't inevitable that it would fall along party lines--would put the Senate in the place of the court system, interpreting case law and making judicial decisions. So, no, impeachment is not and would not be appropriate until we get a final ruling.
Yes, the president can be sued on a personal level. Nixon was sued when he refused to turn over a number of Watergate tapes, and the Supreme Court unanimously ruled against him. I'm sure that's the model Paul is following, for two reasons: 1. The SCOTUS can issue a definitive ruling that the NSA program/activities are unconstitutional, which Congress through an impeachment proceeding cannot, and, 2. The SCOTUS can order injunctive relief, preventing any such programs (at least that they know about) from happening in the future.
The reason Paul comes off as a whiny bitch is--even apart from this being a transparent publicity move with little chance of success--because he's a member of the legislative branch who's gone crying to the judiciary about Obama rather than sacking up, rallying the troops, and doing something about it in Congress.
