I’m using Copay as my Bitcoin wallet. I see it is opensource with the code available in github. What guarantees that they uploaded binary based on that source code? Does it go through Apple approval? Storing more money in bitcoin, I want to be sure my password / credentials are safe and that the developers are not sending out in a hidden way, via backdoor.
You don't have any guarantee. You are relying completely on the developers to submit the binary application file based on that source. You are also relying on Apple not to manipulate the application binary once it gets to them.
It's no different with Bitcoin Core or any other wallet or software. In all cases you're putting your trust in whoever does the build and the source and tools they use.
One option is to look at the history of the software. If it's relatively mature and has had no faults, either malicious or competency related, then it may be safe to put a degree of trust in it.
Would not a paper wallet be considered the safest ? Only you know under which stone you tucked away the note about private key and backup phrase.