I assume you are talking about Kentucky bluegrass seed. It might be dead in June, but who's counting? Now is NOT a good time to put bermuda seed down because you have not had enough hot weather yet. The soil temp must be very warm for the bermuda seed to germinate. Usually for Houston that would be later next month. If you put it down now you would be watering and watering and wondering and wondering. Then, by the middle of next month it would suddenly pop up for you. But in the mean time you wasted all that water.

Take this opportunity to find the Bermuda Bible online and memorize it. It will suggest mowing at a low height. When you mow Kentucky bluegrass that low, the Houston heat will wipe it out pretty fast. No need for any herbicides or anything.

You will never need to cover anything with a sprinkle of topsoil unless you need to permanently change your soil drainage. If you have standing water somewhere when it rains, then bring in the topsoil. Otherwise, leave well enough alone. If you want to cover it with something, use a light dusting of compost. The rate is 1 cubic yard of compost per 1,000 square feet. Compost is different from soil and will not change your drainage.

How often do you want to mow? Once a week comman Bermuda works (just ugle IMHO) twice a week get a good Bermuda seed like Riviera or Yukon. You can still mow these once a week they will just get a layer of thatch and won't look great for a day or so after you mow.you will need to raise mower height once a month. No big deal but these two are much softer.