Is it common for hotels/hostels to need to see your passport when checking in? Also, is it common for transportation services to need to see it or have the number if purchasing online (I.e. when buying bus tickets, train tickets, etc)?
Transportation, beats me, never been there, but giving it out can do you no harm.
Yes. Hotels will ask for your passport when you check in and they might even hold onto it for a little while. Typically they need it to register you with the local police. That's a normal practice throughout Europe. You will also need to show your passport to a conductor with your rail pass or ticket. If you are shopping and request a VAT form they will need your passport to fill out the paperwork. I don't think you should have to enter your passport information on line when buying tickets but you should be prepared to show it to anyone validating tickets or checking them.
E-tickets often require a form of ID, such as a passport or the same credit card used for purchase, to dissuade people photocopying the tickets.