What happens to the ground wire in this case (the outer part of the Coax output)? is the shield of the coax cable acting as the "ground"?
Yes, and so is all other connected metal, possibly including the USB cable and the attached computer. There might or might not be a path to earth there, but I find it better to think about it not as “ground”, but as “the other half of the antenna” — because it all affects the radiation pattern. The point of adding a “ground plane” is not to make the antenna “grounded” but to provide that “other half” in a consistent fashion so the antenna will have predictable behavior.
However, for hobby receiving purposes, there is no reason to worry about using the antenna “correctly”. Just play with it and find what receives your signal better.
if so then does it not matter that the shield is not 180 degrees opposite of the antenna like a dipole is?
It matters but it doesn't make the antenna ineffective. An L-shaped antenna (the cable running sideways out of the antenna base) is still a usable antenna. Picture an antenna with radials at the base.
If you decrease the angle between elements all the way to zero, the antenna would theoretically not receive anything because it is now more like a transmission line (intended not to radiate!) than an antenna. But outside of that perfectly null scenario, there will be reception.
Is there a better way to "ground" it if I don't have access to the ground itself.
The earth itself is actually nearly irrelevant to setting up your antenna unless you are putting it right up against the earth. It's too many wavelengths away to matter, otherwise.
As already mentioned, any large sheet of metal, like a baking pan or a metal cabinet or desk, will work well for this purpose. It should have at least as much radius (length from the antenna base to the edge of the metal) all around the antenna as the length of the antenna rod itself.