Zinc has been proven to help prevent common viral illness, shorten their course, and lessen their symptoms. There is no reason to think it would act differently with COVID, and some speculation I've read says that the anti-malarials are working at least in part due to making zinc more effective.

Vitamin C is controversial. Personally, I think the evidence shows that extra vitamin C has no benefit in preventing or treating acute illness. However, there are some true believers that think Vitamin C is great for anything from colds to life-threatening sepsis. Also note that I said "extra" vitamin C. Most Americans are not vitamin C deficient, but if your diet is particularly strict or particularly bad then you should be taking supplements already.

As far as the dosage, it just depends on the vitamin. RDA for vitamin C is pretty low, so your vitamin likely has plenty extra. Probably not the case for zinc