Most commonly, there can be a Response to Motion for Summary Judgment filed. Motion practice is very intricate with many rules that can hang up even the most experienced attorney. You will probably be well suited to consult with a competent attorney in your area that normally handles cases like yours.
If the adverse party has filed a motion for summary judgment, you can obviously and should file an objection to the motion. You can also file your own motion for summary judgment and argue that there are no genuine issues of material fact and you are entitled to a judgment as a matter of law. You can also file a partial motion for summary judgment. In other words, you can file a motion for summary judgment on liability only.
I agree with the previous commenters that it is most appropriate to include a counter-motion for summary judgment in your opposition to their motion for summary judgment. As to your final questions, usually the scheduling order issued by the court will include a final date by which all "dispositive motions" must be filed. Check this date to determine the last date you can file your motion. Also, whether you can file one or more motions depends on how the court handles your first motion for summary judgment. If they dismiss it with prejudice, you're done. If they dismiss "without prejudice" or with leave to renew the motion, then you can file a subsequent motion.
As stated by previous commenters, I would recommend consulting an attorney. If your case is good it will be well worth the money you spend if it increases your chances of winning on summary judgment. The advice given here is only general advice and should not be relied upon without consulting an attorney.
Best of luck.
You should definitely file an Opposition to the Motion for Summary Judgment and may, if the facts and law support your position, file your own Cross Motion for Summary Judgment. The other party then would file their Opposition to your Cross Motion for Summary Judgment.
I agree with the other commentators that the court will generally set a time limit for dispositive motions to be heard. You should check with the court to see what the docket calendar has for your case.