Maybe, but only Princess will know - have you tried the roll call or the Princess board?
Boots are normally provided at snowmobile safari and dogsledding - normally not for Northern Lights tours - might even not be required.
Might also be late for many activities.
Ålesund and Bergen easy walkable and many sights Geiranger has a Ho Ho bus to the view points. Eidfjord is a very tiny city. Try also Norway Excursions
Might help with a note accurate if you tell which cruise line.
In Europe prices normally does include all fee and tax. Gratitude included in the EU price while not included in the US price- if booking any US price gratitude will be added once in board - at the end prices will be very similar. Again might differ for various cruise lines - I have most experience with Royal Caribbean and Celebrity.