I am not sure what you are referring to?

If you have a mobile pass you need to add every train into your TRIP in the App. Before you board the train you operate the slider and this adds that train to your travel day and records it automatically for inspection on that day.

With a paper pass you write the details of each train into your pass, again for inspection onboard the train.

I am not aware of any other requirement to record your journeys, other than for your own records.

So it is purely for own record and not needed by ticket inspection 

To be able to present a QR Code for ticket inspection, you need to add at least one trip of that day to your journey in the App.

To conform with the Terms of Service of the Pass as I know, you need to add every leg you take to your journey. In theory, that can be checked by ticket inspectors, as the legs are listed below the generated QR-Code. (Thats needed as the payout of the rail companies is based on that data). It is also very nice to rewind the trip later by checking your journey diary.

No. They are an integral part of your ticket. Without them, your pass is not valid.

You have to record every single train In times of Paperpass i saw travlers that had a book full with the train details. 

Because only then you have a valid ticket. The Record of the MobilePasses is now used to divide the money between the Rail & Ferry companies that take part. In the old times with just paperpasses you had to send your pass back to Eurail (After few weeks you got it with a small present back)