1. If the heater windings are not shielded from the high-voltage B+ windings, switch noise from the B+ section can be capacitively coupled to the low-voltage heater windings. Although indirectly heated tubes have significant isolation from line noise, it is not 100%, so some of that switch-buzz gets through. If you listen closely, you will probably hear it.
There are several solutions: A) get a power transformer with electrostatic shielding (copper foil) between HV and LV sections B) go to DC heating with a regulator C) use a separate heater transformer.
When none of these are done, yeah, that’s cost-cutting, no matter how fancy the name brand. If swapping the rectifier tubes makes a notable difference in the sound of the amp, there’s something wrong, or not optimal, with the power supply design.
2. Yes, the two grounds must connect together at one point, and only one point, preferably the chassis. There’s an old joke: when two engineers get together, they will come up with three grounding solutions, and all of them will be right.
Grounds (and shields) should never float, ever. That’s bad practice and a possible safety hazard. Determining the best connection for the ground or shield usually takes a bit of thought (which is why it’s called engineering).
There are whole books on grounding. It is not a trivial subject. Part of the problem is language, because one word covers several things:
A) Safety ground
B) Current return path for the DC currents that power the tubes or transistors
C) Current return path for the AC audio signals going through the circuit
D) RF and noise shielding