What variables affect the height of a Heron's fountain?

[haruspex]

ok. tube f is a tube full of water which connects a water medium to another water medium.

Yes, A and C, not A and B.
Have you lost track of where we are at?
You need to make a list of the pairs of surfaces which are connected by a water only path, and a list of pairs connected by an air only path.
In post #56 you claimed there was such a path connecting A and B. Are you ready to agree there is not one?

 

[Physicist1011]

oh sorry tube d definitely isn't one.

 

[haruspex]

oh sorry tube d definitely isn't one.

Ok.
In post #56, you correctly had A and C connected by water.
For B and C you wrote "same", but it is not clear whether you meant they are the same pressure or that the connection is the same type as AB. Please clarify.
Finally, you need to say what the base of the fountain (the point inside tube d that's at the same height as surface A) connects to, and through what medium.

 

[Physicist1011]

B and C are of the same pressure. The base of the fountain connects from b into the atmosphere - through both water and air.

 

[haruspex]

B and C are of the same pressure.

Right.

The base of the fountain connects from b into the atmosphere - through both water and air.

No, remember that we are supposing you have a finger on top of that tube, so the fountain base is isolated from the atmosphere.
Now, what equations can you write relating the four pressures?