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

We had a similar experience about 30 years ago. Along with that same pine tree we had a huge full blue spruce right next to it in front of our house. Both trees were full with branches right to the ground, healthy & beautiful. It made our house disappear & look like a matchbox house & prevented light to come in. Both were 75' tall & growing nonstop. It started to affect our pipes. I don't like cutting down healthy trees. I began calling some large landscape companies asking them if they'd take them. One company had a customer who wanted live trees as a fence & barrier in an expensive suburb near Boston. They gave me a price & thought it was fair. We were really happy they wouldn't be destroyed. What a day it was when they came to remove those trees! Neighbors came over & cars driving by pulled over to watch. It was fascinating how a huge machine fit around the tree & lifted it up, roots impact & no damage at all. It took two days & the landscaper handed us a check which paid for filling those very large & deep holes. IMO I'd leave the tree for the new owners to decide whether to remove it or not. It's not that uncommon as you'd think.

What about installing some wires/trellis to train potted star jasmine? The benefit there is fast coverage, looks attractive from both sides year round and smells good when in flower. You could put in some simple rectangular pots for the jasmine, along with a few feature pots with other plants Eg. Patio-suitable citrus. Do you have information on climate/personal tastes etc. (A few things to consider... Height from ground/access to trim a creeper such as the jasmine, weight of pots/soil, water run off from watering)

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.