Not really, no, unless the grass is quite thin and the phlox quite healthy. Then anything's possible.

But in a well-maintained lawn, it shouldn't do that.

IME from what I see around me, yes. But if you have the bed edged well and check it periodically, you can keep them separated.

I had this when I lived in Arkansas and Bermuda grass got into it. It was a fatally inseparable mess. Since you mow the lawn, I wouldn't worry near as much about c phlox getting into the grass. But if you have aggressive grass, don't let it into the phlox.

The phlox will creep. It won't spread into your lawn like creeping Charlie or other weeds. If you keep mowing to the edge of the lawn (and over any phlow that crept in) you won't have a problem. The phlox will not survive mowing and the grass will take over.

I love my creeping phlox. It's no trouble keeping it in check.

I am attempting to replace areas with grass, that are unable to be mowed. I want to plant a ground cover that wont erode on a hillside and dam. I mow above and below so It could be kept in check, Except for along the edges that but against wooded area. Should I consider this or something else?