How do different resolutions (pixel sizes) work? I get that everything memory related (including color mode and depth) could be switched on a raster line.

In theory it should be possible to change the frequency of the video signal mid screen as well, but I have a hard time to imagine switching repeatedly every frame wouldn't have driven the monitors of the time crazy. Even multi-sync monitors needed probably a couple of frames to sync, right?

In the context of the Amiga you didn’t need to change the signal frequency, the underlying video signal stayed the same.

For wider pixels, the hardware just drew each pixel for longer, and for taller pixels each pixel spanned more lines.

Pixels weren’t real in the CRT days :)

It’s basically just an SD TV signal, the vertical height in lines doesn’t change, it largely just swaps if each line outputs 320 or 640 pixels during its assigned time period.

Having interlaced and non interlaced modes at once seems a bit funky though, but I assume it just forces everything into interlaced with the non interlaced mode sections having the same line sent on both fields?

Analog monitors don't care about the dot clock. They draw lines, but you can put whatever signal you want in the lines so long as the dot clock doesn't exceed the available bandwidth. So horizontal resolution is effectively a range you can change at any time and the monitor won't even notice.

Vertical resolution is very much part of the spec, but even then CRTs will sync to hilariously out-of-spec signals that gain or lose lines per frame. Sanely-designed graphics hardware like the Amiga wouldn't do this, but the Atari 2600 wasn't sanely designed and had plenty of games that played fast and loose with NTSC. Atari graphics hardware only generated a single line of graphics and relied on H-Blank effects for literally everything else. Even the vertical retrace signal was controlled by the game. So it was very common to see badly programmed games send too many lines, and a different wrong number of lines each frame, which nobody noticed until people started writing 2600 emulators.