100% a job for a professional, a proper assessment of drainage and ground condition would need to be done before you could safely start changing anything here. It’s important to keep in mind that like anything, what goes underneath is usually the most expensive part of a project and the most important, finishing touches with only last of what’s holding them is build to a good standard. As far as materials are concerned that really depends on your budget and how much you want to alter the overall area. Composite decking offers a variety of options with level changes and is easy to maintain, sandstone as mentioned would work very well with the surrounding area and some stone if used correctly would offer help with drainage.

If its a solid floor (assuming it is which is probably where the moisture is coming from) then yeah lifting the flooring and some kind of DPC (there are various kinds) will be the solution, and then some kind of barrier behind skirting I think is a technique to help anything not propagate up walls. But an independent damp surveyor is the way to go, if you get a company round who apply solutions they may say you need more done than you actually do.

Is yours the ground floor flat yeah? Send a pic of the outside of the property in the area of damp. If the render has been done right down to ground level then that's your problem zrender should stop above DPC which is normally about 4 ish bricks above the ground. If your render goes over it then it defeats what the DPC is trying to do (be a damp barrier) as render then acts as bridge for moisture over DPC

It seems like there are two issues- firstly do you take the belt and braces approach of stripping your side roof and making it like new or do you just deal with the immediate issue- I say ask roofer number 3. Separately you need to remove the cause of the problem which is technically a communal issue. I say look at your lease agreement and if time is of the essence you cover yourself with lots of pictures- get the work done at your expense and then pursue your upstairs neighbours for a contribution afterwards.