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A graviton is the smallest possible unit of a gravitational wave. The amplitude of the wave corresponds to the number of gravitons, like you said, and its frequency to their frequency (quantum particles have frequencies that are related to their momenta). We're aware that light, at least, works like that.

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A gravitational wave requires an event like a black hole merger, or basicially somerhing to move and change the field, right?

In this case, how does the fact that a big object is still influencing space/time around it communicate that fact when it is not moving. Is that still gravitrons?

Everything creates gravitational waves. They are very difficult to detect, so we can only detect the ones created by black hole merger or something similar.

Assuming our guess about quantum gravity are correct, the normal gravitation force use gravitons too, they are virtual gravitons but the distinction between "real" and "virtual" particles is another whole can of worms.

Gravitons impacting and imparting momentum seems like it would have a bunch of observational implications. Does a massive object cast a graviton shadow? Is the momentum positive or negative?

Classical waves do all of those things too.

So gravitons are only emitted by accelerating objects?

Real gravitons, yes.

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