Wizz Air's fee table show seat fee to be €4.00 – €60.00, i.e. it depends a lot on route, seat, and potentially timing of booking. I'm not entirely sure about Whizz Air but chances are they deploy "dynamic pricing" i.e. they adjust the price over time based on supply and demand.
Many budget carriers will assign you the worst available seat at check-in to incentivize you to pay for a better one. The cost of that seat will depend on what they perceive is the difference between what you have been assigned and what you can pay for and also how full the flight is and how much seats assignments they think they can still sell.
is the price of buying a seat after random allocation the same as if buying a seat before random allocation?
Quite likely. I've certainly seen examples where the fee after check-in was significantly higher that at booking time. You might get lucky, you might not.
Does the price change over time, e.g. if I leave it too late?
It often does.
Is it possible to buy a seat (after random allocation) online and not queue at the airport check-in?
Yes.
If I check-in at the airport specifically for buying a seat will I have extra costs? e.g. "you did not do an online checkin blahblah"
That's very likely. WhizzAir will charge extra for checking in at the airport even if you don't reserve a seat. That's probably the most expensive choice. You should check-in online if any possible.
There are different strategies to optimize around assigned seats. Ironically, if you are planning to NOT pay for a seat, it may be better to wait to the last minute, because by that time all middle seats may already have been assigned to non-paying passengers.
On the other hand: I once got assigned the middle seat in the back row that no one else wanted and so I ended up with the entire row to myself. YMMV.