What's your question? Ask

Look for a grass specific herbicide, which will not kill perennials and most ground covers.
You can remove the Phlox and treat the grass; then replant. Do this in the fall.

Mulch only lightly (1 or 2 inches), keeping the plants damp and especially removing weeds so they don't spread along with the ground cover. Once you've helped the ground cover fill in, you'll have less and less weeding to do (maybe even none at some point) and no more mulch to add.
https://www.gardeningknowhow.com/ornamental/groundcover/creeping-phlox/creeping-phlox-planting.htm

I would wait until the heat waves of summer are past. It needs full sun to bloom heavily which will be hard for a new transplant to take when temps are high. Make sure you gradually get it used to full sun. If you decide to put it out now, construct some shade for 3-5 days, longer if it struggles in full sun. Water when the soil is dry 1" deep. Make sure it doesn't dry out if fall is dry. It needs to be well-hydrated going into winter.

These will need to be planted about 1 foot apart in any given direction from each other. This article will give you more information on caring for these plants: https://www.gardeningknowhow.com/ornamental/groundcover/creeping-phlox/creeping-phlox-planting.htm

You can bring it inside. There would be no problem with that. Just treat it as you would if it were growing outside, with the exception of watering it much less. Over-watering indoors will lead to rot. This would be best placed in a south facing window, and can stay there until spring.

Your plants look pretty normal for winter.
Cut out any old, woody stems that no longer produce flowers during the later winter pruning, removing these stems at their base. These stems typically form near the center, or older part of the phlox. Summer growth replaces the old stems with tender new stems that will flower.
https://www.gardeningknowhow.com/ornamental/groundcover/creeping-phlox/creeping-phlox-planting.htm