Specified in the footprint.
The hole diameter specified in kicad is typically the final hole size as you receive the board.
This size depends on the size of the component lead. (Rule of thumb: 0.2mm larger diameter compared to the maximum lead diameter)
Specified in the footprint.
The pad diameter is normally quite a bit larger than the hole to allow for soldering. There even is a minimum size increase specified by your manufacturer. (minimum annular ring.)
clearance can be specified on the pad, footprint or project level.
(pad setting overwrites footprint which in-turn overwrites project settings. Setting the pad or footprint settings to 0 means take the one of the level above.)
You only get an error if the other trace or pad is assigned to a different net.
And yes this clearance is defined on a net by net basis via the netclass. But it can also be specified at the pad or footprint level.
More details see: Tutorial: How to make a footprint in KiCad 5.1.x (From scratch)?