Looks like buffalo in the second lot of photos but it also looks like you could have some couch in there too Buffalo is a great wearing lawn but it is hard and scratchy and kids don't like playing on it too much, unless it is Sir Walter, which is a more friendlier version and the most popular one now. It will respond well to a sandy type of top dressing - black river sand is really good if you can stand the mess for a few weeks while it settles - a sandy soil will give the runners something to grab onto and grow and this is what buffalo needs. I would leave the couch in there for now but dig it out as the buffalo runs to achieve a really even lawn. The buffalo will probably choke most of it out too but do try to control weeds early as it is harder to get them out when the lawn is established and many weed killers will kill buffalo too. Buffalo is very much a surface grass so the clay won't bother it but it will affect your lawn in terms of run off and water absorbency. With winter now approaching, there won't be a lot you can do but study run off over the coming months and where the water pools, time it takes the grass to dry out after rain and then decide just how much you need to do to control water problems. Maybe just running some agriculture piping around the edge of the paving will solve your problems or there may be other areas some piping can help drain quickly. Nothing worse or more irritating than water problems, so it is really worth the effort fixing them in the coming months ready for next spring, the problem will be fixed forever and you can focus on letting the lawn grow. There are quite a few websites with tips on growing buffalo and problems you need to watch out for. This one gives good tips on mowing heights too, which is very important, especially in your heat. If you mow it too low the sun can burn it and kill it. http://softleafbuffalograss.com.au/buffalo-lawn-care-and-articles/maintaining-buffalo-lawn.php

Congrats on choosing blocks for your garden bed. If you're looking for trees, I suggest considering a maple tree. They grow nicely over time, providing shade and privacy. I recently moved to a new house myself, and I'm also exploring plant options. Back in my hometown, we had a reliable tree service that helped us with planting and maintenance. Maybe you can find something similar in your area. I came across this website, https://www.deepdale-trees.co.uk , which might be helpful for you. They offer a variety of trees and services.

Great ideas dreamer,a dark colour fence highlights plants beautifully, you could always have a feature to look out on using screens , you could add a vertical garden , standard bay trees either side would look gorgeous, even a small water feature bowl , you can get them in terracotta would look beautiful.

looks like one of the Philodendrons or similar plants that are usually grown in pots as indoor plants but anything growing in brickwork needs to be removed to avoid damage to the wall