For metric threads, coarse is considered 'normal' and where it isn't specified, you can generally assume it means coarse pitch. For reference, for M4 this would be a pitch of 0.7mm
From my understanding HQS is a carbon steel tap, cheaper and (in theory) won't last as long and HSS is more expensive and you could expect to maybe last longer. I would suspect that HSS would be more brittle and probably wouldn't handle as much mistreatment (but I may have that back to front) either way, in these sizes, niether will be particularly robust.
If you are just rethreading a brass insert then a HQS plug/bottoming tap would do the job, no need to start with a taper tap.