Great question, and thanks, Peter, for your exhaustive answer!
As a Slav, I can confirm that the term is, indeed, Slavic. "Molieben" (with soft "l") is the way Russians pronounce it; "moleben' " (with hard "e" after hard "l" and with softened last "n") is more of a Ukrainian pronounciation. Etimologically, it is related to the noun "molytva" (prayer) and to the verb "molytysya" (to pray). It is a church service specifically asked or solicited by a certain member or members of the parish, either to commemorate a certain event and to praise God for it (for example, a moleben' to praise God and thank Him on the occasion of winning the lottery
) , or a moleben' to pray God to prevent danger (for example, a moleben' for a safe journey.)
Panakhyda (Rus. "panikhida") is, indeed, very different from a moleben'. It is strictly a service for the dead. The main part of it is singing "Memory Eternal" ("Vyechnaja pamiat' " in Russian, "vichnaja pam"jat' " in Ukrainian). It's long and extremely solemn.