Is a galaxy wide language possible?
I would say that this is not really possible unless faster-than-light communication is possible. We currently know of no way to do this, so barring some breakthrough in physics a galactic language is highly unlikely. This is even the case if the galaxy is populated by a single species.
Why? The milky way galaxy is about 100,000 light years wide.
Consider the following - in an attempt to establish a galactic language, a galactic radio station is set up somewhere near the center of the galaxy and broadcasts to the entire galaxy. However, the broadcast is going to take 50,000 years to reach the edges of the galaxy.
So in order to have a truly galactic language, language is going to have to be static across the galaxy. However, language shifts with culture, so culture would have to be mostly stagnant. A stagnant culture isn't good for stability though - the people on the bottom are going to want to move to the top, and the social convectional currents caused by that movement stirs things up enough to cause cultural changes. So in order to have a stagnant culture the people on the bottom need to stay on the bottom, and there's little chance that could last 50,000 without revolts.
What about a non-constant galactic language?
With no FTL travel possible, not being able to have a true galactic language doesn't actually matter. Suppose it's accepted that the language of the galactic radio station is the common language to use. Then you can tune into the galactic radio while you are traveling to another planet. Given that star systems are on the scale of a couple light-years apart, it's going to take you at least a couple years to get there. That's plenty of time to pick up the galactic language even if you hadn't known it before.
Even if the galactic language were naturally changing, there wouldn't be enough change to matter in a local setting. The language near the center of the galaxy would likely be very different from what it is around the edge of the galaxy, but if an immortal or long-lived being were to travel from the edge of the galaxy to the center of the galaxy it would take 100,000 years in a very fast space ship (traveling at half the speed of light). As they listened to the radio over the course of the 100,000 years, they would hear the language go through 150,000 years of evolution (remember that the edge starts of 50,000 years behind the center). I'm sure you could keep up with the evolution of a language that changes only 50% faster than normal.
What about relativistic effects? If you're traveling fast enough that reaching the center of the galaxy will only take a few years from your reference frame, you'll still need to accelerate and slow down. The time you take decelerating would be a perfect opportunity to catch up on the changes in the galactic language.
What about different species?
Our galactic radio station actually suggests a rather simple solution for this! A radio turns electromagnetic waves into sound waves that we can hear, so why wouldn't we expect it to be able to do the same for other species‐it turns electromagnetic waves into signals they can understand. So all you need to communicate with someone is basically a walkie-talkie—your device knows how to convert your sounds into the proper electromagnetic waves and vice versa, and their device know how to do the same for their sounds.
In this sense, neither you nor the being you'd be talking to would be speaking the galactic language. The true galactic language would be the electromagnetic one broadcast by the galactic radio station.