As for why Wormtongue was needed with the King being ensorcelled like that, I think it was basically to keep someone like Theodred or Eomer from taking control and making Rohan an effective foe of Saruman despite the loss of the King. It was important that Rohan had NO King, not just the shutting down of Theoden.
As for the tear in the movie, it is hard to say. It seems out of character to be anything aside from a tear of pure joy at the fact that success is staring him in the face. Then again, perhaps he had suddenly realized that things have gone too far, and that the kingdom of Men he thought to rule was probably going to be wiped out entirely.
It would be interesting to see what Grima's actor would say on the matter in the audio commentary on this movie's expanded DVD. Having all the "new crew" (ie: those who weren't in "Fellowship") do the commentary itself, perhaps with a couple of others, would be even better!