Well that sheds some new light on this one.
The round casting that the seat is bolted to is most likely the 3 point cylinder casting. (on my Jinma, that is inside the hydraulic tank so it looks considerably different)
With the seat flipped forward, it appears that there is a breather tube between the springs. What breaths through this tube is still unclear.
The sump that is bolted to the side of the transmission is what we first thought was a filter housing, but you have a spin on filter hanging under the starter so the first part must really be a sump.
Have you unscrewed the cap to see what is inside? When looking inside you should be able to tell if it is a stand alone container or if it has a hole to let fluid into the transmission.
Does this tractor have power steering?
There is a brass fitting with splines on it just under the front edge of the seat. Undoubtedly there was once a plastic knob molded onto this fitting. That is the knob that controls how fast the 3 point lowers. If you turn that knob which is actually a needle valve all the way down, it will prevent the 3point from lowering at all. Assuming no leaks internally.
The transmission and hydraulics may indeed share the same fluid, but I am not yet convinced of that.
The picture from underneath suggests that the sump is much smaller than I originally thought so it may just be a fill point.
Please pull the cap off of the sump and show us a picture of what is inside. If full of fluid you could take a piece of wire and bend about an inch at 90 degrees. Then used that to feel around inside to see if you can find an opening into the transmission.
Keep digging, you will figure it out.