As booking.com states, requirements made directly to the hotel, like extra beds, cots for babies or pets allowed at extra pay, will be charged by the hotel.
Those are charges you should know about as you requested them or agreed to them.
Tourist tax is also to be paid directly to the hotel but is often already mentioned on the booking page.
An extra charge ...
It really depends and even Booking.com does not know which is why they use the word may. Generally speaking there are additional fees that they either cannot collect or cannot calculate due to complex rules that need more information than provided.
Sometimes there are no additional fees at all but often there is something charged by the locality. From the ...
In almost all cases the answer is yes. The check in agent has full access to the ticket details including the fare class and the specific conditions & restrictions. Otherwise, they wouldn't be able to do any type of changes, upgrades, re-bookings, etc.