No price drops inside final payment unless booked within the past day. Also, some of the prices you are seeing for your category may be for new bookings only.
Same price on board but even then, don't buy it until you ask the barista how many punches for your drink - the math for some drinks doesn't make sense with the card.
You can go the most out of the way bathroom on the island but it doesn't mean it will be any different than the rest. There's quite a bit of subjectivity to "clean" when it comes to toilets.
Likely this but it won't hurt to ask - guest services. A cabin may not become available until after sail away (no show), which will be after you have already settled into your new cabin.
You might think that but here are the numbers from the Q3 filing last year when a bunch of ships were in Europe. The numbers don't change that much with the varied home porting of ships.