It’s true that Russia is a member of the Council of Europe and currently has a moratorium on the death penalty (though it remains on the statute books). However it has no bid to join the EU and thus Russian laws or court verdicts cannot be challenged or affected in any way by Brussels (can you REALLY imagine the Russians standing for that???). Trials by jury were also largely abolished by Putin in 2008:
Well, we are both right and wrong. I have now checked and it was just an Italian PM invitation for Putin to join, not their own bid.
As for European Human Right Court it has power over Russian national judicial bodies due to Russian membership in a Counsil of Europe.