The pixels are not exactly square, the actual aspect ratio depends on the TV system.
For PAL-B, the pixel aspect ratio is 0.937:1, the pixels are a bit higher than they are wide.
For NTSC, the ratio is 0.75:1. At this ratio it can be clearly seen that the cursor block is definitely higher than wide.
In standard bitmap mode the C64 outputs 320 pixels in 40µs.
The visible portion of a line is ~52µs; in 60Hz regions ~240 lines are considered 'visible', but in PAL regions it's ~288 lines.
So if there were no borders, there'd be around 52/40*320 = 416 pixels across the visible portion of a line.
Given that each line is 4/3rds as wide as a hypothetical ...
As for the first part of your question, yes, for sprites as well as characters or bitmap images the same principle holds: either you can be in single colour/hi-res mode (320x200, 1 bit of foreground colour info per logical pixel) or multi colour mode (160x200, 2 bits of colour info per logical pixel).
In practice, for PAL, a single colour pixel that is ...