do ADRs pay dividends?

yes, they can. depends on the company.

are there ever any cases where a stock pays dividends in their home country, but not in the states? how would the value of the stock differ?

all Indian companies treat the ADRs as extensions of their local stock and dividends are converted into dollars and paid.

I haven’t seen an ADR that pays a dividend in its home country not pay one in the ADR. You may have to pay a bit of a tax/fee on the dividend as it comes in however, so a .60 dividend may have an alteration of a few cents. Not a big deal for some people, but instituitions are typically better off buying it in the local market.