Would graphite stick to basalt cooled from its molten state in a graphite mold?
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.