Is it possible to get a Sleep Apnea study in Germany?

Your primary doctor can refer you to Landstuhl for further assessment/studies.
This will not change your deployable status unless you are found unfit (untreatable condition) and then you are referred for a MED, per AR 40-501. It would be a process, check out the regulation and speak with your provider.

Absolutely get the sleep study done. Apnea takes a good 7 years off your life left untreated. Lots of health issues, i.e. blood pressure, and others go along with it. Typically you're deployable so you get to lug your CPAP along. Companies make camo cases for them no less. You'll be able to dream again once you start the REM sleep. The diagnosis is also important for when you get out as it typically rates 50% disability on its own. BTW, you get pretty creative with CPAPs as I go camping or whatever with one. A 12V deep cycle battery will run it for at least a week if you're not using the humidifier. There's also a study method where they send you a monitor you put on yourself, sleep a night, and send it back. It all depends on what the pulmonologist wants to do.

I'm stationed in Kaiserslautern and had the sleep study conducted about a year ago. My advice, for what it's worth, get it done. Anytime I deploy, go TDY, take leave, my machine goes on every trip with me and my wife is thankful. There is a bit of a process to get the study conducted, but it's well worth it, and the CVR (CPAP supply company) are very accommodating.