Of course, you can do that. Seems like a good idea if you don't get distracted from your work too much.

I probably would bring a hard drive though, not drop box it. DV files are compressed very little and will therefore become huge (I remember it being about 1GB per 5 minutes video, which was ridiculously much back in the days... but for internet transfer, it is still a lot).
Whatever program you use, it will most likely capture the original DV compressed data from your camcorder. So the video quality won't differ between different programs and you can use whatever you like. The only difference will be the container it is packed into. And there might be some additional work here.
The first two tools that I found that can capture DV for Windows (never done it myself) are
Windows Media Encoder, which should be pre-installed, and a little tool called
WinDV. The latter will write out an .AVI container. The Windows Media Encoder in its newest version might give you several options. If it does, try to get in descending order: MOV, DV, AVI, WMV. Most likely, MOV will sadly not be on the list, but one can dream...
If you bring whatever file you end up with to your Mac and it won't import in FCE, don't worry, this is just because of the container, not the codec, and there is no time-consuming and quality-reducing re-encoding necessary. Just open the file in
MPEG Streamclip, chose "File->Save As" and save it in another container format. It shouldn't convert and therefore not take a lot of time.
Good luck!
PS: If the footage was shot in 16:9 and looks like 4:3 in FCE, this can happen. Just ask again.