When bench testing a chassis does the flyback and the socket connected to do the tests?

The yoke needs to be connected, and the flyback does if the rest of the horizontal section (output tube, etc.) is complete, but the HV cable to the CRT can usually be left disconnected (but you should insulate it or put it into a glass jar for safety).

Need to keep the B+ loaded to keep it from going too high, so yes, should keep the sweeps running for trouble shooting. The horizontal and high voltage particularly depend on having the yoke connected.

For Tuner/IF alignment, though, the voltage pulses and ground currents from the sweeps may disturb the scope waveforms; so some chassis alignment instructions include disabling the sweeps and using a specific large power resistor to load the B+.

Ok thanks that answered some of my questions!

If it is not a transformer power supply type chassis then disconnecting the CRT usually disconnects power from all(or many) of the tube heaters(there are workarounds for that though).

If it is a black and white set and you plan on working on others like it then getting a test CRT (5AXP4 and 8AXP4 are fairly common/universal types) would probably be a good investment(you can often get one for under 15$ on Ebay).