"How can I be sure, that all of MY notes in Evernote are being synced correctly?"

"I'd rather use a service where I don't need to double-check if my data is synced or not"

Which is it? Do you care, or don't you?

Your best bet is to distribute the syncing/backup. Use the primary service for syncing, and an alternate service to backup the data the applications utilize. CrashPlan fulfills the second part of that for me.

Sure. But all software has bugs.

Your only real solution here is to not rely on one service or one company's engineering. Had I been in OP's position, I'd have an rsync script set up on a server somewhere copying the important note somewhere else daily. Perhaps to a dedicated space on my personal machine, where I edit it. The script would perhaps send me an email or notification whenever it runs so I don't have to check it manually.

In practice I don't have single points of lifestyle failure. I learned my lesson after I locked myself out of my Gmail account out of stupidity. I was able to regain access through a nifty little trick, the account I'd set up to recover it from I'd stopped using a long time ago, but I was able to re-register it for the sole purpose of recovering my Gmail. Now I use LastPass but I have my Gmail password memorized. If I lose access to LastPass it's not the end of the world, I can still reset all my passwords.

You want rock-solid reliability without having to engineer it yourself? You're dreaming. The only service tailor-made for your needs is the one you build.