Welcome to the forum. Directly to your questions:
1) As you say, I cannot make a specific diagnosis. From your description, I doubt your penile lesion was a wart; it sounds more like a skin tag. If it persists after the cryotherapy, perhaps a biopsy would make sense; this is something to discuss with the dermatologist.
2) Cryotherapy kills all tissues, so no matter what the cause, the lesion probably will be gone. If it recurs, see no. 1 above.
3) There is nothing you can do. There are are no nutritional or vitamin supplements known to have any effect on clearance of warts or other HPV lesions, or skin tags.
4) I recommend you clarify with the dermatologist whether s/he thought the lesion was a wart. If it was, you should discuss it with your partner and make a joint decision about sexual plans. The thread linked below, and several other threads linked there, go into great detail about various aspects of genital HPV infection that you may wish to consider. The main point is that warts and most other HPV infections are best viewed as minor inconveniences, not important health risks. Many couples decide to not worry about special measures to prevent transmission.
http://www.medhelp.org/posts/STDs/HPV-concerns/show/1819004
I hope this has been helpful. Best wishes-- HHH, MD