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You can go ahead and cut them back anytime after flowering has stopped. Deadheading them may even promote additional flowering. Here is more information: https://www.gardeningknowhow.com/ornamental/flowers/fgen/deadheading-flowers.htm
https://www.gardeningknowhow.com/ornamental/flowers/phlox/growing-garden-phlox.htm

I would guess that the nitrogen to phosphorus ratio is off. The soil may have an abundance of nitrogen, which would result in nice, healthy green growth but little to no blooms. Without the right amount of phosphorus, it will not bloom well. Bone meal is a great way to add phosphorus.

The stems are normally cut back after flowering to promote a second bloom. Care of creeping phlox also includes cutting the plant back in late winter/early spring prior to new growth to allow for rejuvenation and to produce young, more compact stems.

I would let them be right now. If you cut the plant back now, it will only encourage new growth, which is not a good idea this time of year as the cold weather soon to come will only damage the plant, which should actually be getting ready to go dormant for the season.

Yes, you can cut the spent flower stems to the ground at any time, but you should generally wait until spring for pruning back the remainder of the plant, just prior to new growth, as this normally triggers the regrowth process. If it is ginning to look unsightly, however, it will not hurt to give them a light trim.

Unfortunately, there are no selective herbicides that will affect only broom sage. The only option would be to hand pull the weed out.

After planting, new growth typically appears in 2-3 weeks depending on soil and air temperature. So, it's a bit too soon to tell yet, I would give it some more time.

For more information on phlox, please visit the following link:
https://www.gardeningknowhow.com/ornamental/flowers/phlox/growing-garden-phlox.htm