Variation in collector current if we change the collector resistor in a common emitter transistor
To a first approximation, the transistor is a current source, so changing the collector resistor does not change the current, only the collector voltage

We can model the transistor as either a base current controlled current source (hFE, beta), or a base voltage controlled current source (gm). Which model is most convenient depends on whether we're biassing the transistor, or using it as a small signal amplifier.

To a higher order approximation, the collector current will change slightly with the collector voltage. You've already mentioned the Early effect, so you understand that.

For very high values of Rc, where V+/Rc exceeds the current the transistor is trying to sink, the transistor will be saturated, and neither collector voltage nor current will change.