The amp would in effect add mass to the cab, but how much this affects the bass response would depend on how rigid and heavy the cab is in the first place. Mesa cabs (for example) are pretty hefty to start with, less beefy cabs might show more of a difference.
No, but if I move the aerial around it does.
Since tubes are microphonic, amp sitting on the cab would transmit more vibrations and colour the tone more. Having the amp in a different room altogether would go the other way. Some individual tubes will be more microphonic than others, and the design and construction of the amp is a factor.
YMMV depending on your gear, but IME the audible effects are subtle at the most.
P.S.
Now I've mentioned aerials it reminds me that moving the amp away from the speaker will reduce the influence of any electromagnetic noise coupling from the amp to the speaker coils, and reduce HF coupling due to stray capacitance also. These are electromagnetic compatibility issues, which should be negligible if your amp was carefully designed with respect to EMC. Which isn't a given.