Right, the focal length remains the same no matter what size the sensor is.
The effective focal length however is based on sensor size, it's called crop factor, but that was implied in the OP's question initially, so I didn't find any reason to repeat it.
F-stop is based on the effective focal length, as I said, the aperture diameter is fixed, however effective focal length is not, even though the physical focal length is always fixed.
But crop factor is an abstraction, it's a way to calculate how to get a certain result. So yes, crop factor includes changes to F-stop, for-instance:
Sigma 18-35mm F/1.8 is the physical dimensions and obviously can't be changed.
However it's an APS-C sensor only lens, so the 35mm equivalence is approximately:
Sigma 27-50mm F/2.8
The DoF is determined by the range to subject. When increasing the distance, because of the crop factor on any given focal length, then the DoF is increased accordingly. The total amount of light gathered by the sensor for the same exposure time and aperture, is also less by the same crop factor.
F-stop (not aperture) needs crop factor applied as well. Hope I have express myself more clearly now.
And the downvotes, only shows, some are into alternative facts. Those seeking the truth, will however try it out and perhaps like those links.
Considering all in all, I'll consider the downvotes medals of honour, the definitive proof I hit a nerve.
Videos and links to support the fact, can be seen here: