ElizabethB at Helpful Gardener:
Mint is a real beast and will survive inspite of anything you do except over watering.
Mint seems to be "all or nothing" (somewhat like Marmite's "You either love it or you hate it"). Usually "all" (too much growth), though a few people have had no success with it. Last week I had more than a square metre of it removed from where it had escaped its pot (planted outside). It was rampant but had never received any special attention (no fertiliser, no watering whatever the weather, except some of the spray intended for the nearby vegetables in times of drought).
Mint roots are not aerial and should be spreading out to seek nutrients and moisture for the plant and for mint that means underground. They do grow very shallow but actually on the surface seems odd and it could be that those attempting to grow underground have rotted. You might find the entire plant comes away from its compost with just a gentle tug.
As ElizabethB said "The only way you can kill mint is to love it to death." So I suggest a dose of neglect. Don't water it again for at least a week and if it then perks up a little still hold off watering until it starts to show signs of thirst. Then start a watering regimen at a slight greater frequency. Do prefer rain water, or if tap water at least let that stand overnight before adding it.
As a back up I suggest taking a cutting (eg) even if shorter than the 8cm recommended there and, in due course, use potting compost rather than garden soil.
But to answer your questions (!):
Is it safe to change the soil completely for mint?
Yes. In a pot potting compost would be better than garden soil.
And is it bad for the plants that the roots are sticking out?
No, but in my opinion a sign that something is not right.