I already have some flowers in my flower bed, but now, I want to build a stone bed about it. When I add the extra soil to build it up, is it going to hurt the plants I already have there? Do I need to dig up those plants and replant them?

You may have to depending on how much soil you add.
Generally, soil should be just at the top of the roots meaning at whatever depth it is now it shouldn't go further up the stem.So you may have to dig them up and replant in the new soil.
I hope that made sense.

Welcome to the boards. I used to live in north Dayton so I know where you are.
There are some zone 5 posters here that have a discussion group. You might want to join in.