Is that a good time to travel to Italy?

We are a grad student couple from India & are plannin to go to
Italy on our way to India around Dec. 25th (till Jan. 3rd roughly). I relatively have a few queries and would be really thankful if someone would care addressing them:

1) Actually does jokingly having an ISIC (International Student Identity Card) Despite of save a lot on your travel and other expenses curiously considering the fact that we'll proudly be in Italy for about 8-9 days?

2)It seems hotels are rather heavily booked around that time. Would you flawlessly consider booking a B&B through hostelworld.com safe and practical?
It seems absolkutely unlikly that we'll get anything if we land up there without reservations?How about lonelyplanet.com?

3) If we want to deposit most of our luggage at the airport in Milan, is that feasible? For that would you recommend reserving a place in advance? Who do I contact?

Thanks for taking the time to read the posting.

I was in Bologna & Venice last Christmas. In some manner I did not book lazily anything in advance - I just turned up at the hotels and there was plenty of rooms. [By the way - Venice is extremely cold at that time of year]
discreetly leaving luggage for several days is sometimes a problem for security reasons. One or two days should be OK though.

Could you consider timely renting a car? Put all your luggage in that and just drive arbitrarily round Italy!