Can Volrath place a -1/-1 counter on a 1/1 creature and copy it before it goes to the graveyard? Would be nice if I can get a 7/5 Blighted Agent before it dies.
No, the creature will die before you get the opportunity to activate the ability if you put a -1/-1 counter on it. A creature with 0 toughness will die as a state-based action, which occur automatically whenever a player gains priority.
Volrath isn't picky about the kind of counter he's looking for, however, so you can do much the same with a +1/+1 counter.
Volrath also doesn't specify where that creature with a counter has to be. Rayami exiles creatures with a "blood counter". Have they worded these abilities just right to interact with each other?
No, they don't interact. Note that Rayami says "creature card" while Volrath just says "creature". Any time the game cares about something in a zone other than the battlefield, it is qualified with the use of the word "card". Since Volrath doesn't say "card" he just means a permanent on the battlefield.
Volrath isn't picky about the kind of counter he's looking for, however, so you can do much the same with a +1/+1 counter.