Would graphite stick to basalt cooled from its molten state in a graphite mold? Molten metal that is poured into a graphite mold pops out easily once it has cooled. Would basalt poured the same way pop out just as easily? 

I am particularly interested in lunar basalt, if that makes any difference. The image shows the composition used to make a lunar regolith simulant, taken from chapter 13 of Resources of Near Earth Space, by B. J. Pletka.
No, it would not stick. In our lab we commonly use graphite dishes/cups/etc to contain small amounts of liquid rock. All you need to do then is to hold it upside down and the solidified rock (now glass mostly) falls out. Sometimes there's a bit of graphite dust on it, but it can be easily scraped off.

Doing it in a reducing (or oxygen-free) environment will make it even easier, because graphite slowly burns to carbon dioxide with time, on the scale of hours, from my own personal experience.