Since there is limited space, a simple rectangle would seem the most efficient use of it ... the size being according to the number of users, furniture and functions.

Around the perimeter, screening could be done by using a line of repeating small tree forms. You only need them to be 10' or 12' high in order to gain privacy and screen out the objectionable view. They could all be in a strip bed of flowering groundcover so as not to be individually mowed around.

This is a good start! I love the tree idea and how the branches stay above the tree line so it doesn’t make the yard feel even smaller. I have a dog that enjoys eating everything that fits in her mouth. Would you recommend a container garden as opposed to a bed? What would be the best way to hide and soundproof the noisy ac unit?

Personally, I wouldn't do a container garden because it's more maintenance and more difficult to keep plants going ... to no great advantage. And it's more expensive per square foot.

Plants won't quiet the AC unit. For that, you'll need to need to create a solid panel -- enough distance away not to impede the air flow. Solid masonry works best but often isn't practical. A solid wood panel is next best.

Oh wow this looks great Doug! What would be the recommended height and dimensions of the deck?

With limited space it will be difficult to block the view of the house. But once you give the eyes something else to focus on the view of the house will become secondary. Think about a pretty landscape bed near the seating area incorporating a water feature, a bird bath or feeder. A wind activated spinner would also draw the eyes as well. You can’t hide an elephant in your yard but you can draw attention away from it

I would suggest having the patio come out approx. 20'.

The house along your back fence offers you one thing, privacy for your large window. You could have a two story house in your backyard looking right into your yard and windows. I would do a lovely planting bed, something easy to maintain, lovely to look at, dog friendly, a small tree for height (maybe planted in front of one of the house’s 2 windows) and call it a day.