Achievable Peak Height with a Water Pump - Fountain vs. Hose 
As shown in the picture, I have a small water pump with a round outflow of diameter d that generates a fountain of height k. I now want to connect a water hose to this pump to drive water over an obstacle (the flow rate is not important for the application).


*

*Can we estimate the achievable height h from k?

*If not, is it possible with additional parameters? 

*If not, can we at least say that one is always larger than the other?

*Can we optimize h, for example by using a hose with a large or small diameter?

 A: 
Can we estimate the achievable height h from k?

The achievable height is not determined by the fountain height 'k'. For example, consider a piston pump which slowly pumps water at high pressure to its output. It wouldn't be able to achieve a very high fountain height 'k' because the rate at which pumps water is so slow, but since it can pump water against a very large back pressure it would be able to pump water to a large height 'h'. 

If not, is it possible with additional parameters?

The achievable height 'h' is determined by how high of a back pressure the pump can work against. If that information is known, then the height 'h' can be determined.

If not, can we at least say that one is always larger than the other?

Obviously, the maximum achievable height 'h' has to be at least as large as the fountain height 'k' since one can always take a fountain of water and then just enclose the fountain of water in a hose. 

Can we optimize h, for example by using a hose with a large or small
  diameter?

For an ideal pump, I believe that the height 'h' is determined only by how much back pressure the pump can work against. In the real world, though, the height may be also be affected by the hose diameter - or at least hose diameters above a certain size - because larger hose diameters mean that the pump initially has to work at pushing a larger volume (and, consequently, weight) of water over the wall, and that may conflict with the design limits of the pump.
