I need to block out my neighbor's yard from my dogs. There is about ten feet of fence with about another ten feet of fence with the privacy slats. It is on the driveway side, so there is no grass. This is a rental for a short period, so spending a lot is senseless. I was thinking of string beans. I have done morning glories, but the bees were a problem and the plants didn't do as well as expected. This is a VERY sunny area.
Containers are an effective way to create privacy. These can consist of rows or layers. Many small trees and shrubs are well suited for potted environments. Alternatively, you could also choose a variety of tall-growing grasses, bamboos, and vines. This article should help: https://www.gardeningknowhow.com/special/spaces/plant-a-privacy-screen-plants-that-grow-fast-for-privacy.htm
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