I don't have a great deal of experience with this, but my understanding is the following:
If it's a U.S. listed foreign corporation?
Typically the company that pays a dividend pays the dividend in local currency for their local listing, then converts the dividend to the foreign currency for holders of their foreign listed shares at a specific rate or the prevailing rate on a particular day. E.g. if the company pays the dividend in yen, the company will convert the yen to USD on a particular day and pay that amount to US shareholders.
If it's an ADR?
The operators of the ADR will convert the dividend to the foreign currency and pay the ADR holders the converted amount. The operator of the ADR should be able to clarify the specifics if they are not made clear in their prospectus.
Is there an implicit "pari passu" clause? Can they pay less dividend to U.S. investors than Japanese investor?
While I do not think there is a law that requires it, it would make the US shares less desirable (and therefore less liquid) than Japanese shares, potentially undermining the potential for raising additional capital through the additional listing. It would also open an arbitrage opportunity where US investors could short the US listed shares and buy the Japanese shares for a risk free profit.
What is the exchange rate for converting dividend? As of which date?
The company with the foreign listing may announce the rate or the date at the same time they announce the dividend. The ADR operator may have their own methodology. The company should provide this information in the prospectus for their foreign listing, and the ADR operator should provide this information in their prospectus. If not, their investor relations department should be able to provide this information.