Since the Universe is expanding, is it accurate to say that a galaxy is 5 billion light years away? Isn't it better to say that the light from the galaxy took 5 billion light years to reach us and it is a very old picture of it? However, if light travelled back to the galaxy at the constant speed and an observer was stationed to record it wouldn't it take longer to reach because of the expansion of the Universe? 

Also if it still exists after 10 billion years or not we could not predict. So is the picture of the Universe more of a timeline of events rather than like a conventional map that shows distance between points.

How do current day theories tackle these issues?
I'm not 100% sure if I'm understanding what your asking, but if your question could be rephrased as "how do we measure distances in an expanding universe?", then I can try to answer that.

Depending on what astronomers measure they use different distance measurements. For example the comoving distance between two objects takes into account the expansion, and so does not change with time. If you know the redshift of the galaxy, for example by measuring the spectrum, and have a cosmological model, then you can calculate the comoving distance. Here cosmological model means constraints on the amount of dark energy, matter (both dark and regular) and radiation. The current accepted model is that the universe is around 70% dark energy and 30% matter, most of which is dark (with negligible radiation). The percent here refers to the fraction of energy density. Note that these values change over time, mostly since dark energy is like a property of space and so increases as the universe expands.

For some more info see:
https://en.m.wikipedia.org/wiki/Distance_measures_(cosmology)

Note that light years is a definite measurement, and so we can use it a valid unit for all distance measures.