First of all the hydro fluid has water in it. Not good. It all depends what you want to invest in this machine . The wheels have seals that will need fixed. If I were going to keep it I would get all the hydro fluid out and clean and dry the tank. Fix the wheel seal. Then find out where the engine is leaking. Could be valve cover or the main seals. You may need a service manual to refer to. I have repaired and used two skids and you have a lot of time and money invested. Parts can be hard to get. I make all the gaskets I can. Only when you have repairs made and all new fluids and filters replaced do you run it. Good luck.
Good question.... If it leaks out of all it's orifices, you are kinda wasting your time giving the system a real good clean just yet.
I'd start by pumping all the oil out you can then re-filling with fresh cheap oil, be it engine or hydraulic oil. Run the machine a bit and dump it again, re-fill then start chasing your leaks.
I'd start at one spot and fix the leak, move on and on till they are all clean.
Once you have most of the oily bits done, you can then concentrate on more oil flushes.
IF you are planning on a full strip of the machine to do the leaks, there is no point flushing the oil, as you will have all the parts off and you can blow them out with compressed air.
I have re-built machine from the ground up, so all the oil was removed and the system was squeaky clean.