Is it legal for a third party in general to stamp my passport?
Your passport won't be stamped. Only the print out of your boarding pass, with a stamp that says "VISA VERIFIED" and a signature. Here's an example:
Is this a one-time document check only?
No, unfortunately you will have to go through the same procedure every time you fly. Just why Ryanair cares about checking visas on intra-Schengen flights while full-fare airlines don't care is a mystery.
Does this mean I cannot check-in online?
You can check-in online. In fact you definitely should unless you're willing to pay €35 for checking in at the airport.
The only difference is that you will have to visit the Ryanair check-in counter at least 45 minutes before your flight to stamp your boarding pass. The check-in counter is the same place where you drop-off your luggage if you have one.