I live in Clarksville, TN and I need to find a climbing flower that does great in full shade for my trellis. I would love to find something yellow or red.
You might like a trumpet vine for that area:
http://www.fcps.edu/islandcreekes/ecology/trumpet_creeper.htm
https://www.gardeningknowhow.com/ornamental/vines/trumpet-vine/growing-trumpet-vines.htm
Most places have it listed as partial shade, but I have seen it grow fine in full shade. It will not grow as fast, but this is not a bad thing with trumpet vines.
I have a half wine barrel as a pot in a cottage-feel garden and am wondering what would be a good feature tree?
Many people like Japanese maples. This article has some others that are popular: https://www.gardeningknowhow.com/special/containers/how-to-grow-container-trees.htm
What types of plants look best to create a hedge around the house?
If you are looking for simply a decorative shrub, boxwoods are a popular choice. If you are looking for privacy, these articles will help:
https://www.gardeningknowhow.com/special/spaces/plant-a-privacy-screen-plants-that-grow-fast-for-privacy.htm
https://www.gardeningknowhow.com/special/spaces/creative-screening-with-plants-good-borders-make-good-neighbors.htm
What is best to use as edging between a gravel driveway and a semi-circular lawn?
I personally prefer low growing, hardier sedums for spaces like that. They take a lot of abuse, spread well, and keep grass at bay but don't compete with grass.
I live in Galveston, TX and the wind out here is crazy! Are there any flowers or shrubs (other than hibiscus) that I can plant? How about a lemon or lime tree?
This article has some good suggestions for a windy garden:
https://www.gardeningknowhow.com/special/spaces/wind-resistant-plants-for-your-windy-garden.htm
I am not sure how a citrus tree would do. Winds can be very drying and citrus plants need their moisture to grow well.
We're in need of a fast-growing (up to 15 feet or so) evergreen privacy hedge that can be planted close to our house and fence. The house will only be a few feet from the fence/hedge/tree screen, so we do not want to plant anything that has invasive roots that may compromise our home's foundation/structure. Any ideas on what will work? It's partial sun/shade and a tight spot.
The following articles should be of some help to you:
https://www.gardeningknowhow.com/special/spaces/plant-a-privacy-screen-plants-that-grow-fast-for-privacy.htm
https://www.gardeningknowhow.com/special/spaces/creative-screening-with-plants-good-borders-make-good-neighbors.htm
I would like this evergreen shrub to be flowering, if possible. We are in Hendersonville, in western North Carolina.
This article will help you:
https://www.gardeningknowhow.com/special/shade/evergreens-for-shade.htm
Mountain Laurel is a popular flowering shade shrub in your area.