I have to put in a new laminate floor. There are already a set of sliding mirror doors in the closet. The closet doors have a bottom track in which the doors click into to move smoothly. I'm not sure how to do the flooring around the track. Should I remove the track and lay the flooring underneath the track and then screw the track into the flooring once it's in? Should I cut the flooring so it butts up against the track, at which point the track would be slightly lower than the flooring?

The closet has a cheap outdoor carpet on the closet floor that I stapled in. Another option I was thinking was that I remove the track, put another piece of the outdoor carpet underneath the track and then screw the track in on top of this new piece of carpet.

Just looking for any suggestions on how to do this. I've looked online and searched for advice but everything I find includes videos on how to lay down flooring, which after the 5th video, just reiterates the same thing.

Not quite sure I understand your question... But Will try.
If this is a floating floor it must have room to expand, this means no nails or screws should be used.
It this floor is stapled or nailed, then the track can be laid on top.

Basically, my laminate flooring is going to be a floating floor install. My closet doors have a bottom track. What do people do when the flooring butts up against the bottom track of the closet doors? If I cut the laminate flooring with a circular saw, it's going to look like an unfinished edge on the flooring.

Post a picture.
Is this a bifold door, or a bipass door?
I've never seen either with a track at the bottom.
May have to go with a trasition stip on the laminite side of the door.