Certainly. If the motion is granted, it resolves those issues in the case (and may resolve the entire case) eliminating the need to have a trial on those issues.
The other party will not do a motion to dismiss the motion for summary judgement. The other party will either (1) agree that there are no material facts in dispute but make a cross motion for summary judgment arguing that the law favors him rather than the other side or (2) oppose the summary judgment and ask the judge to deny it, asserting that one or more material facts are in dispute and thus a trial is needed to resolve the dispute on those facts.
Yes, unless the parties settle it before trial, of course.
Not usually. Only if the motion is clearly frivolous or otherwise violates the rules would the party seeking summary judgment face some kind of sanction by the judge. Motions for summary judgement are very, very common.

