GP are not the best sources of information on this. Sperm analysis should be one of the first things done because it's easy and noninvasive.
But, it's your guys' decision to make and i"ll quit harping on it.
I personally do not recommend multivitamin for you, but if you want to, I'd do it only 3x a week. DH can do it daily. Multivitamins for men will have adequate zinc in them no need for more for him.
If you have been going on on strict diet with few cheats for longer than 6 months you should add in 8-15 mg zinc 3x a week for you. If you do the multi 3x a week it will have that zinc in it and you don't need to add the extra zinc in that case.