Hi, I've checked the Dutch website of P&O Ferries:
https://www.poferries.com/nl/veerboten-naar-engeland

and all it says is that "don't forget to attach your bike well to your car".

It also gives a link with details https://www.poferries.com/nl/onze-routes/calais-naar-dover/reisinformatie/reizen-met-een-fiets - but it only brings you to contact details of their support (klantenservice@poferries.com, etc).

As to your hotel question - for me, as someone with a Dutch passport, it's hard to imagine a bike-unfriendly hotel in Netherlands (unless it's some sort of seedy downtown hostel oriented towards rather different to cyclists  tourist groups, or something so small that they have no lockable fietsenstalling (i.e. bicycle shed).

(dictionary: fiets(en) - bicycle(s); stalling - shed/shelter/racks)

You'd be certainly fine with one of local hotel chains, such as Van der Valk Hotels.

Definitely used to be possible. In fact the older booking system fpr P and O only allowed you to book 1 vehicle and did allow bicycle to be selected. If you tried to book 2 people a warning used to pop up saying that you needed to book through the booking office instead "unless your bike was a tandem" which I thought was great. :-D

When I looked last summer the website had changed. Now the bicycle option only comes up if you have one person - it means that single cyclists can book on line but if you are 2 or more cyclists wanting to be on the same booking I assume you still have to call the booking office.

https://www.poferries.com/en/routes/hull-to-rotterdam/travel-information/travelling-with-a-bicycle

At the bottom of the page it gives you the booking office number.

It's a shame it no longer defaults to allowing 2 people bookings and it being a tandem though. :-D

Also tells you that you check in through the car lanes and walk or cycle up the ramp - you don't check in as a foot passenger.

Unfortunately there is now no mention of tandems on the website (it was only that pop up when booking before) but I think it will still be possible. Best to call the booking office to check.

Yep, although this was a few years back. We parked up in the car park at Hull (Quite expensive!), then rode onto the ferry. Warning, if you arrive on a bike, you are expected to cycle, not walk, onto the ferry. It's a very steep ramp too and we had an impatient driver right behind us all the way up. We didn't have to book a bike back then, we just booked as foot passengers. Once on the ferry we lashed the bike to the central barrier with some of the ropes that were already there. Getting off was much easier, they let the cyclists off first so we rolled down the ramp, first through the customs office and straight onto a cycle path. It's about 20 miles to Rotterdam. Going there we went on a lovely, scenic route. Coming back we used a more direct route through an industrial area. Can't help with hotels in those areas I'm afraid. We arrived early morning and rode all the way to Woerden (65 miles) on the first day. Blog linked below.

https://samuriinbred.wordpress.com/tandeming-the-netherlands/

I've used the ferry a few times and previously have used one of the on-line parking sites such as parkonmydrive or just park.  Significantly cheaper than the multistory but last time I parked very close to the docks and when I returned the car was covered in guano so to speak.  If using the ferry again I would park nearer the university.  

This year I'll be using DFDS from Newcastle to Amsterdam, the ferry is fractionally cheaper and it may suit you depending on where you're travelling from.  If I remember correctly P&O charge the tandem as 2 solo bikes, as does DFDS.

As for places to stay, also consider B&Bs, I've found them to be great value for money and very welcomingly.  I usually use Booking.com to make the reservation and add a comment regarding tandem storage.  

I lived in Rotterdam (Wester part) for ~10 years, and cycled a lot around. One of my frequent routes was to Hoek van Holland - that's where the Harwich Stena Line ferry goes -  and back.

If I were you, I'd avoid going to Rotterdam, but instead either go South-West: Oostvoorne-Rockanje, then, roughly, N57, along the coast - it's soon  becomes a separate multi-lane bicycle "highway", there kilometer-long beaches, small old fishing towns, etc etc)

Or cross the canal to Rotterdam on a local car/bicycle veerboot (ferry) -  very frequent, Rozenburg-Maassluis, and follow the coast North to The Hague, Katwijk, and then Amsterdam. You can also deviate to Delft.

Also, Maasluis-Rottedam has a nice cycling path all the way to Rotterdam, if you must - nicer than staying on the Southern shore of the canal with all its industry.