Merete and Co :-),
As someone else mentioned, your post would have been better in the "To The Boot" section, as this section is for reviews.
A few thoughts regarding your questions.....
- YES, it's definitely a good idea to pre-book hotel rooms, especially in May as it will be starting to get busy.
- Where to Stay? Everyone has their own preference, and my favourite is Monterosso. It's the largest of the five towns and therefore has the greatest number of lodgings, restaurants, etc. For a group of five, that will provide more choices in accommodations.
- Private Rooms. Some of the smaller towns have more private rooms than conventional "hotels", so that will be the only option in some places.
- YES, it's absolutely the best idea to stay in one town and use that as a "base" to explore the others. It's VERY easy to get between the towns using the frequent and cheap local trains (DON'T forget to validate!). You can also hike between some towns (depending on the status of the No. 2 trails in May) and take the train when you get tired of hiking.
- Accommodation suggestions. As it's not likely that you have a copy of the RS Italy Guidebook at hand (which has lots of hotel listings), these two websites may be helpful:
http://www.rebuildmonterosso.com/p/visitors-information.html
http://savevernazza.com/visiting-vernazza/
I'm not sure whether the lists are completely up-to-date, but you can check various hotels to determine room availability. My favourite hotel is Hotel Villa Steno in Monterosso, but you could also have a look at Hotel Pasquale or Villa Adriana (or any of the others on the list for Monterosso). I'm not as familiar with Vernazza and the other towns, but I'm sure others here will be able to provide some answers.
Happy travels!