Is liquid hydrogen rocket propeller pollution free?
I'll address you questions in order:  

1) Yes, that is basically the correct reaction, although not quite balanced;  here is how I would write it:  

$$\ce{2H2(l) + O2(l) -&gt; 2H_2O(g)}$$  

2) Ideally, if there were no side reactions, then yes, this particular reaction is "polution free", and the only product is water.  

3) You are correct in your thinking that in the end this does not represent a net pollution-free energy source.  

The reason for this doesn't lie in the reaction taking place at the rocket, but in how they produced the hydrogen in the first place. Unfortunately, there are no significant deposits of $\ce{H2}$ on our planet waiting to be pumped out and used as fuel. So we  have to make it, which requires engergy.  

This means that the $\ce{H2}$ is not a net energy source, rather it is essentially acting as a battery, storing the energy from some other source. So, you can think of the actual pollutants of the $\ce{H2}$ rocket fuel as being the products from whatever process went into generating it in the first place.  This largely depends on what part of the world the hydrogen generation took place. They could have burned coal or other fossil fuel to produce the energy, or they could have built a dam to generate hydroelectric power, etc.  Any of the negative aspects of producing a particular type of energy would be considered a negative product of the $\ce{H2}$ production.