How to determine the theoretical carbonic acid concentration in soda water from a Soda Stream
The equilibrium you describe is quite well documented (especially with entire industries depending on their ability to carbonate drinks), so the Wikipedia page for carbonic acid is often a good place to start. The equilibrium constant, ${K_\mathrm{h}}$, is given by:

$$\ce{\frac{[H_2CO_3]}{[CO_2]}}=\pu{1.7e-3}$$

Now, for your first question:


  How can one determine the gaseous $\ce{CO_2}$ concentration knowing only the temperature (assume room temperature) and the pressure (above).


Well, for starters you should probably check the volume of the container and from there you could work out the amount of $\ce{CO2}$ that is above the solution. But if you are interested in the amount inside solution, you could check this paper published in 1940 which has basically compiled the data for you.

And the second part:


  How can one determine the carbonic acid concentration knowing only the dissolved $\ce{CO2(g)}$ concentration?


Now that you know the equilibrium constant, I'm sure you can work out the amount of carbonic acid.