It's not impossible at all. Russians flying to Turkey for a vacation is still normal - yes, even ordinary middle class people. My friends living in EU are using this to meet with their family in safety.

That's very inaccurate. The German case states that the possibility to be drafted alone doesn't constitute grounds for asylum, you need some form individual persecution (because of ethnicity, gender, sexuality, religion etc) in Germany. "I got charged because my daughter drew a picture that called for peace" will be a very different case than "I'd rather not fight personally".

It's not new either, it has pretty much always been that way. A lot of German politicians are calling for exceptions for Russians (which would effectively end the practice for all countries) to weaken Russia's ability to draft more people.

I think you can easily fly to turkey and then anywhere else. Would not even be that expensive to get from istanbul to sofia..

With regards to visa some friends applied for student visa and that seems to work quite ok.

Moscow - Astana flight with luggage is ~230 USD, railroad is even cheaper, that's about all the additional expenses. The visas are the main issue.