I would ask a JD dealer what fluid. I know it will probably be $10 a quart. Use a turkey baster to fill fluid. I would not risk using just any fluid in it yet.
The hose by the battery? If it is clear and 1/4" hose, it is probably the battery vent hose. The hydro vent hose will be a black 3/8" hose. Yours might have the vent directly on trans under fan covered in dry grass.
If they stop working when warm, it usually means the hydro seals/pistons are worn. You might get lucky and be low on fluid. I just checked a diagram of the stx38 black deck hydro. It does have pistons but is not the same as the link above. There is a filler plug(#2) right rear of transaxle body on top.