How safe? It is not safe if you store them as plaintext.
You can verify this for yourself. Build a sample Corona app for iOS and, from the command line, cd into the .app directory. Run strings on the resource.car file. I did this on Corona’s “fishies” app and found, in part, this:
bubble\_strong\_wav.wav viewableContentWidth viewableContentHeight container newRect reflectX newImage aquariumbackgroundIPhone.jpg aquariumbackgroundIPhoneLandscape.jpg
That’s for data in string format. Now the lua bytecode is going to be pretty painful to parse by hand, though I wouldn’t be surprised if someone had written a script to reverse it.
So if you want to hide secret values, then you should obfuscate. This is a technique that goes back at least to the 1980s, copy protection, 5.25" floppies, extra tracks, etc. Barring a TPM (which we’re not talking about here), obfuscation is doomed against a sufficiently determined attacker, so you need to hope that your attacker is insufficiently determined.
Others may have poked around inside the innards of Corona-built apps more than I have and I’d be interested to hear details if anyone has them.
The question about remote server safety and security is rather open-ended and too broad in scope for me to answer.