Is it "safe" to provide users with the apk file? I was wondering, if I have an app that is available on the Play Store, is it "safe" to provide a direct download of the apk file to someone (either signed with the proper keys, or just debug-version)?

"safe" in the sense of does the marketplace do anything special to the apk to protect the source code, signing keys, etc.?

Reason I ask is a user of my app has said that their Marketplace/Play Store is no longer working and therefore can not upgrade the app - so asked me to email them the apk file.

The app is a free app, so on the surface I can't see any issues - though just wanted to double check.

Although this raises the question - the apks of the apps you download from the marketplace (paid and unpaid) are on your phone. Are these apks (on the phone) somehow unique to the phone/google account? Otherwise, what stops one from buying an app and passing that apk on to others?
There is nothing dangerous about it, and this is done occasionally when the store isn't accessible.

In response to your second question, apks are not unique - and can be easily transferred to another device, or the internet. This is the base of android app piracy.

When you upload the APK to Google Play, GPlay does nothing to it. It is up to the developer to obfuscate the code, to prevent decompilation.