I am skeptical that you have molluscum in all the spots you've mentioned. No doubt you had them to start with, but by now you may be over-interpreting small skin spots like blocked follicles that aren't mollusca. Before you make any more health and life decisions about these spots, you need to check them out by seeing a doctor with a fresh pair of eyes.
1) Is there a treatment for M.C. lesions on the lips?
Yes. Freezing, Aldara cream. But I am not at all sure these are mollusca.
2) Can I spread M.C. to a partner through kissing? (having lesions on my lips)
Possible, I guess, but again, these may not be mollusca.
3) I have had M.C. for over two years, will my body eventually mount a response and rid itself of the virus?
Yes, mollusca go away in 12-18 months. If they are really mollusca.
4) If my body mounts a response to the virus, is there a possibility of me passing it down to my children in the future?
5) If I touch a lesion on one part of my body, and touch another part of my body will the virus spread to that part? What is autoinnoculation?
It's unlikely for that to happen, even if they are mollusca, which I doubt.
6) It seems that some of the lesions in the groin area do not respond to the freezing. They get smaler but do not go away, does this mean that freezing is inneffective or that the treatment needs to be stronger?
It may mean that they aren't mollusca.