HI hostels belong to a network called Hosteling International. All HI hostels have to conform to certain specifications, and meet HI standards. To stay at an HI hostel, you can either join HI for a fee and carry one of their cards, or pay a bit extra per night. I think once you stay at two or three HI hostels you become a member automatically, because of the extra money you paid. Some people like HI hostels because you are sure of getting a basic level of comfort, as they tend to all offer the same services. However, they tend not to have any extras. Private hostels can be more of a risk, since they are not inspected, etc., but they also often offer more amenities.
To answer your fist question, I would book your hostels now, as it is so much nicer to arrive in a city already knowing where you are going to spend the night. Good hostels tend to fill fast. In Rome, I second the Beehive - it was great, but it only has eight beds, so it might be booked. In Paris, I stayed at Le Village, but I would not stay there again, as it wasn't the cleanest, and it is not in a very nice area.
Have fun!