Yep, it might be a pain to find a quadruple room in Paris although they do exist. Inexpensive might mean different things to different people, but here are some reasonably priced mid-range hotels with quads:
Hotel Taylor, in the Republique area: http://paris-hotel-taylor.com/
Hotel Choiseul Opera: http://hotelchoiseuloperaparis.com/
Hotel Ascot Opera: http://ascotparishotel.com/
As for getting to Italy, I would fly. I took the train once, and it was quite an adventure, but flying is more comfortable, faster and probably cheaper.
These are the budget airlines I usually check for European destinations:
Wizz Air: http://wizzair.com/
Air Berlin: http://www.airberlin.com/
Make sure to comply with the bag size and numbers, it's taken very seriously.