Which international payment solutions are available for cross-border use?

Have a look at Adyen, they will do what you are looking at, i.e. marketplace payments, on a global scale, whilst still accepting local payment methods.

International Payment Services - Adyen

Refunds will happen mostly on the quoted FX rate for the day if the processor has to buy FX and do the reversal. The loss (if any) for the reversal would be yours to absorb.