Firstly your lawn. It would appear that Taylor Wimpey have laid drainage under your back garden. In doing so the backfill of the trench has not been properly compacted (it never is unless it is under a road or drive!). As a result it has compacted itself overtime and wet weather. This has resulted in slight subsidence. The manhole cover is now "raised" above the surrounding grass areas because that has stayed in the same place. Taylor Wimpey should deposit extra topsoil to the sunken areas and re lay the turf.
As for the waterlogging , this is a common problem that builders can and do wriggle out of. Unless the waterlogged area is within 3 metres of your home then builders are not required to do anything although some do. It seems strange that the sub strata is suitable for a soakaway (this may be under the manhole cover) yet the garden is waterlogged. It seems obvious that the problem could be solved by adding a French drain at the lower end of the slope connected to the soakaway or to free draining ground below.
I suspect that your problems are in the main, caused by a poor quality "topsoil" which has a high clay content. I would get it tested.
The green stain on your brickwork is indeed due to leaking guttering or rainwater downpipe.
It is amazing it has taken this long before anything is being done about it.
The saturated brick wall could and probably will, cause damp and mould issues inside the home too.
Taylor Wimpey should fix the guttering leak. House builders are also responsible for rectifying damage caused by the defect too under the warranty and so will clean the green off, probably with a power washer, although it should also be treated with anti mould treatment too.
Now the hard part...getting TaylorWimpey to put these things right.
If you have any problems contact the CEO Peter Redfern directly:
peter.redfern@taylorwimpey.com He has undertaken to make customer care a priority in the firm's 2015 Annual Report!
Time will tell!