I'm planning to finally climb Mt. Fuji this year before they close it off for winter. I don't really care about seeing the sunrise; my only concern with time is making it on time to the train to go home so I'd like to descend before too late.

How much money does it cost to climb itself (excluding transportation from home to the nearest train station)? Is entry/climbing itself free?
How long does it generally take to climb?
Should you rest in one of the lodges over night to split up the climb? Or is it doable in one go? How much does the overnight stay cost if you do?
How physically demanding is it? I haven't exercised a lot. I'm in ok condition but should I do some cardio or exercise first? My gf suggested maybe she wants to go to, but she fatigues very easily (getting to Disneyland and walking for like an hour makes her need to stop and rest). I'm struggling with if I should tell her to just not go.
How much does the temperature drop? Should I bring like a winter coat and shit even if its hot as balls at sea level?