score:2
(I found it strange as those requirements usually are from RyanAir, and WizzAir is much more flexible, but anyway):
What happens if I use a mobile boarding pass? How do you stamp those? Is Mobile boarding pass even available in Gdansk airport?
In this case you will NOT get a mobile boarding pass. You will be allowed to complete online checkin, but then asked to print the boarding pass (sent to your email or downloaded from website). RyanAir is doing this for those flights.
How strictly is it enforced?
It is not enforced. I have flown tons of those flights last two years, and never did the dedicated document check on either airline. What happens then is one of three things:
It is also my opinion (which some people disagree with) that if you are denied boarding in this case, you shall be eligible for compensation according to EU 261. But I never had a chance to test this yet.
However, I land in the same airport a few days earlier so I figured, would it be possible to have it checked and stamped in advance?
I tried this two years ago, and they refused to stamp it even one day in advance - they said this could only be done once your flight opens for airport check in. And this could be done only at a place where they offer check in (i.e. not ticket sales counter).
Did they give me that answer just so I pay the fee or is it really a big deal?
They gave you that answer so you just paid the fee (or didn't understand the issue). As far as I remember, WizzAir only asks for First/Last name and gender. There is no space to enter your middle name at all.