Traveling from Paris to Lucca by train?

Dear Friends,
I will be traveling from Paris to Lucca by train on 6th June. I will be coming back from Lucca on 14th June and want to stay a night in Pisa and then finally back to Paris. Can you kindly give some information on following matters?
1.What is the best site to book train ticket (passes) online? Can I pay by credit card and submit the online ticket in train? How do I know which route needs prior reservation?
2.Does the train from Paris to Lucca go via Pisa or it goes as Paris- Lucca and finally to Pisa? When coming back, I am planning to travel Lucca to Pisa. Stay one day and then Pisa to Paris. Are trains available from Pisa to Paris?
3.How long does it take to travel from France to Pisa?
4.What is the cheap hotel in Pisa? I will be happy to stay in backpacker if available or in any student accommodation. Basically I want to stay one day in Pisa to visit leaning tower of Pisa.
Thank you.
Neutya.

I would take these trains:
Leave Paris-Bercy station at 18:59 on the night train to Florenze/Firenze Campo di Marte, arrive at 7:16 next a.m.,
then change to a regional train direction "Prato" at 7:49 to Firenze Rifredi at 7:58, (just another station in Florence)
then change to the train to Viareggio at 8:15, which arrives in Lucca at 9:30.

From Lucca to Pisa, there are trains every 30 minutes which take from 28 to 30 minutes. No problem.

From Pisa to Paris:
Leave Pisa Centrale at 19:09 on the Regional train to
Milano Centrale, arrive at 23:15 then change to
EN 220 EuroNight Stendhal to: Paris-Bercy, departs Milano at 23:35, and arrives the next morning at 08:47

The overnight trains will require a reservation.
the regional trains from Lucca to Pisa and Pisa to Milano don't.

You can buy tickets online at http://www.raileurope.com
but they might not be as cheap as on the French rail website (which is complicated to use) http://www.sncf.com

Since the train from Paris to Florence goes through switzerland, you might want to get a Eurail selectpass with 3 countries (FR,IT,SW). You'll still have to pay a supplement for a couchette, but the trains that go through Switz. are expensive.
http://www.raileurope.com/us/rail/passes/eurail_selectpass.htm
You could take a night train from Paris to Ventimiglia, then go down the coast of Italy through Genova, and Pisa, but you'd leave Paris at 9:17 pm, get to Ventimiglia at 8:59 am, then not get to Lucca til after 5 pm. That would be cheaper than going through Switz. but as you see takes a lot longer.

Search here for cheap hotels

http://www.jetsetgotravels.com

It provides search option like near a landmark etc
Also it provide searches from more than one source.

But beware ... cheap hotels can be trouble some times so check reviews of the hotel your are going to book.

Check the exact location of hotel from google maps.