Does the US export the Mark 77 aerial bomb?
I cannot find any info on the export status of the  Mark 77 with a routine search, and plenty of reports of media attention to its use, which leads me to believe there is not a huge market for it among US allies due to the potential for negative press.

For your second question, the Russian equivalent are the ZAB-xxx series of bombs, where the acronym stands for incendiary airdropped bomb (Zazhigahtel'naya Aviabomba) and the number is the warhead weight in kilograms. It seems the most used version at the moment is the ZAB-500. Variations include the OZAB and OFZAB, where the O is for fragmentation (Oskolochnaya) and the F for high-explosive (Fugasnaya). I have not been able to find a link with info on the  OZAB version.



Similar munitions sometimes come with an RBK (Razovaya Bombovaya Kasseta) prefix, meaning cluster dispenser, for example RBK-500-ZAB, which is a standardized RBK-500 cluster dispenser with over 100 ZAB-2.5 incendiary sub-munitions, for an approximate 500kg of combined warhead mass.