Is it a way to restore private key from 12-phrase? I can not restore my wallet from Copay even though I have the 12-word. Is it a way to convert back 'manually' with an other external solution and get the private key? Is the 12-word key storage a universal thing, or only Copay, Bitpay uses it?
You can restore your wallet using another external solution called Exodus.

You can find a step by step process here: How do I restore from my 12 Word Phrase?

I hope it helps you.

There are some other ways as well, I will keep this answer updated.