Italy in December

You have five cities (3 major ones for sights) and two regions on your list, and 16-17 days.

You would rethink this list, perhaps hold it to Venezia, Firenze, Roma and Napoli. You can daytrip to Pisa, and the Vatican is inside Roma.

In winter, hours are shorter, and holidays are December 24-25-26 and January 1. Some things will be closed one or more of those days, but there will be some things to see as well.

You would rethink this list, perhaps hold it to Venezia, Firenze, Roma
and Napoli. You can daytrip to Pisa, and the Vatican is inside Roma.

I will echo this advice. You have too many destinations on your list for 16-17 days and should eliminate the outliers (Corsica and Apulia). Many tourist services will be closed for winter in Corsica, and Apulia is a region - not a single destination - that you would need more time than you have to explore.

Whats the best and cheap way to travel inter city?

Between Rome, Venice, Florence, Naples and Pisa (and the majority of other Italian cities) trains are the best option. If you can nail down an itinerary in advance, you can often land some good deals on super-ecomony (non-changeable, non-refundable) tickets.

Is it easy to hire car/motorcycle there?

I don't recommend a car for the cities listed above or for traveling between them. There are many areas which are entirely pedestrianized and/or which tourists are not allowed to access with a vehicle: there are large fines for doing that. For the same reasons, I would not recommend renting a motorcycle and definitely not unless you are already an experienced operator. All of the cities are easy to get around on foot or via public transit.

Hope hotels and restaurants will be open during that time?

in the cities, yes but expect many restaurant closures Christmas Eve, Christmas Day and Jan 1.

Do people prefer couch surfing?

We personally do not, and I don't expect you're going to find many among Rick Steves posters who do.

We were in Tuscany in December 2015. I think it's a wonderful time to visit. There were Christmas markets, pretty Christmas decorations, street vendors selling roasted chestnuts and much smaller crowds.

It was not crazy cold and wet. There was only one day with off and on drizzle in the afternoon. Many days had plenty of sunshine. There was frost a few mornings, but it warmed up to 50's during the day.

I agree that your schedule is too aggressive. Don't run yourself ragged.

A note on restaurants: many are open and many have special fixed price meals on the 24th, 25th, 31st and 1st. Make reservations a couple of weeks in advance for those lunches or dinners. Stuck with 3 locations: Venice, Rome (and Vatican) and Florence as they have lots to see and do, weather will not impact you as much, they are easy to travel to, and a great first trip itinerary. You will lose some time in sightseeing due to closures for the holidays so more concentrated time is required.
Use the search bar above to look for prior posts on Christmas in Italy or Christmas in Rome. Google those same search terms as well for some useful info on the wider web.

Puglia is a better destination May through October and although I have not been there, so is Corsica I dare say.