I can't offer much advice as I've not had this specific problem. One of the few things that hasn't happened to my home.
As someone who's had several roofing issues, however, I hate water problems. HATE. I feel your pain...
I'm not an expert in this by ANY means. I'm not the handiest with home repairs either, so take my opinions very lightly. I'm sure others with far more experience will chime in.
The options that the basement people recommended sound reasonable to me. I'd think that the important thing would be to figure out how the water is getting in. Not knowing the topography and layout of your home, it's hard to say. If there's a large standing pool of water on the other side of that door, and your basement is lower than that pool, that's a problem as the water will find a way in. IMHO you need to figure out whether it's seeping ground water or if it's a drainage issue. Drainage might be solvable by simply checking your gutters. Maybe one has separated from the house and letting watter pour against the foundation. Or full of leaves and twigs, as you mentioned. Something simple like that. Seepage might be more difficult to track down, if it's truly an issue of ground water seeping in.
Mold can be a bad thing. You've got to be really sure that you get everything as dry as possible. Otherwise you'll never get rid of the 'mildewey' smell. Worse, there can be health concerns as well. At the very least, I'd do a really thourough shampoo job on the entire area. Rent one of those powerful rug-doctor machines if you don't own a good steamer vac. I don't know if you'd really have to replace all of it or not. If you can get it clean & dry in a reasonable amount of time my guess would be that you'd be ok, so long as it doesn't become a chronic problem.
I don't know what to say about the hardwood flooring. Mold is also an issue there but I'd think that warping would be a problem too. Is there any way you can pull up a strip or two to check for moisture? Easier said than done, I imagine. My fear would be that a thin layer of water could get trapped under the boards and spread out over a huge area. Leaving that damp is a welcome invitation for future problems with warping, mold, maybe even termites.
Seriously, good luck. You'll figure it out. These sorts of things always cause me a lot of stress and grief when they happen. But I have learned that it will all be ok in the end. You just have to keep a cool head and have some patience.
