I live in the Houston area. My bottle brush tree was hit hard by a winter freeze and all the leaves are completely brown. Do I prune all the dead stuff, or will the dead leaves fall off when new growth starts (if not completely dead)?
If the stems are still alive, they will regrow their leaves. I would check the stems and if they are still alive, care for the plants as you normally would, removing the brown leaves, and you should see leaves start to grow in a few weeks. These articles will help you as well: https://www.gardeningknowhow.com/plant-problems/environmental/how-to-tell-if-a-plant-is-dead-and-how-to-recover-an-almost-dead-plant.htm
I live in the Texas Gulfcoast area. I planted a Bottlebrush tree in early spring. When I purchased it, it had a few blooms. I planted it in full sun and in healthy soil. Although it continues to grow very well and appears to be very healthy, it will not bloom. Why do I not have any blooms, and what can I do to make it bloom?
If they are getting enough light (which is the most common reasons for plants failing to flower), then it is likely a lack of phosphorus. Plants need phosphorus to flower. Bone meal is a good source of phosphorus (https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/bone-meal-fertilizer.htm).
Do you prune back before where the flower bloomed so that you don't have a bare place on the limb later? Or do you let it sprout on top of the flower and then only prune back an inch or two of new growth?
Prune off new growth as the flowers are finishing their bloom, taking care not to cut into the established portion of the plant with little foliage. Alternately, prune growth from the previous season during the dormant season prior to new growth in the spring.
Over the winter it has gone brown. Is it the frost and will it benefit from cutting back or is it dead?
Yes, this is most likely due to cold damage and the brown/dead growth can normally be trimmed away for new growth to come back. These articles should help as well: https://www.gardeningknowhow.com/plant-problems/environmental/how-to-tell-if-a-plant-is-dead-and-how-to-recover-an-almost-dead-plant.htm
Someone told me to remove the dead flowers by twisting them off and then you don't need to cut or prune if shape is ok. Is this ok to do?
You can cut off the spent flower, that's okay. But pruning the bottle brush is a bit more complicated than that. Here are a couple of articles that tell you more: https://www.gardeningknowhow.com/ornamental/shrubs/bottlebrush/pruning-bottlebrush-plants.htm
I have had this tree for about two years and have noticed the dead limbs. What should I do? Also, when is the best time to feed the bottle brush? What is the best nutrients for the bottle brush tree? I live in the deep south on the Gulf of Mexico. It is hot. Very short mild winters.
These articles should help you determine how to prune and care for your bottle brush tree:
Our bottle brush shrub appears to have winter kill, as the leaves have turned brown. It has always produced blooms and is a relatively small upright shrub. Do I prune heavily or lightly to see if it responds? This shrub faces east and is located in the foundation plantings. All the leaves have turned brown, which I have never before seen on this shrub. It is about 6 feet tall, but not heavily branched. It also experienced an ice storm we had the previous winter (which is very rare here in southern NC); however, the shrub rebounded. We did have an exceptionally cold winter here this year. Should I be wrapping this shrub in burlap each winter? This shrub is about 6-7 years old but has grown slowly.
Yes, we did have an incredibly cold winter, and it would sound as if your Bottlebrush has suffered winter damage. I would recommend pruning the damaged and dead leaves and branches, and wait and see. Give the roots time to wake up and then you can access the plant.
Burlap wrap is a good way to protect shrubs and plantings in the winter.
I have listed a few links for you with more information.