There is no onboard entertainment on Ryanair. There is no point whatsoever in downloading the app, as you will not be able to use mobile boarding passes.
As a non-EU/EEA passport holder, you must print your boarding pass on paper [having it issued at a check-in desk at the airport attracts a fee, unless you are a Flexi Plus customer, and similarly having a boarding pass reissued at the airport attracts a fee for all customers] and the Doc Check desk will only stamp your printed boarding pass (A4, one-sided print only).
Your suggestion above that the "stamp" could be placed in your passport or on an otherwise blank piece of paper makes no sense. The idea of the stamp is to show the gate agent boarding your flight that you have been cleared by Ryanair to fly the sector for which you offer the boarding pass, and every time you fly you must obtain a new doc check stamp for each new boarding pass. If you show a boarding pass without the stamp physically on it, then showing them a Ryanair Doc Check stamp on any other piece of paper means that you have not followed procedure and have not had your boarding pass for that sector checked. Ryanair staff should never place the stamp on anything other than a valid Ryanair boarding pass, so being able to show a stamp on any other piece of paper will mark you out as suspicious. You will not be boarded and will be unlikely to go landside to have your boarding pass stamped and get back airside in time to take your flight.