Well, let's see
Connect to an external webserver:
Being able to connect to a webserver (unless it somehow requires specific applications, anything like that?) is something that any browser can do. Both Mandriva and Fedora come with their share of browser options, I'm sure. So that shouldn't be a problem. And if you're ssh'ing into the server to do maintenance or anything like that, you're still covered. The ssh client is sort of a linux built-in at this point.
Client & Server remote desktop tool:
There are a myriad of remote desktop applications knocking about, VNC being probably the most popular (but there are others). Google would probably be of help in researching this. However - if you'd like to be able to connect to your current desktop from a remote computer, there's pretty much just one vnc server that will do it afaik:
x11vnc. The funky thing with linux and desktops is that the X server (what controls the pretty graphical desktop, all of that stuff) is perfectly happy to just up and create a completely clean, new desktop every time it receives a connection. Which means that ordinarily you wouldn't be able to connect to an existing desktop when you tried to VNC in...you'd get a brand-spanking new one. So if you wanted to log in and check your AIM messages or something, you'd be out of luck. x11vnc can connect to an existing desktop, though...give it a look-see and see if it's what you'd want. The configuration *really* isn't hard.
Gnome and KDE:
Both of them are what's called
Window Managers. Basically this means that while X is perfectly capable of drawing pretty pictures in rough box-like shapes, it does not by default control the interact-able properties of those boxes. Without a windowmanager, windows don't have borders, minimize/maximize/close buttons, aren't movable, things like that. MS windows has the same thing - the window manager is explorer.exe
You just don't hear much about it because you don't have a choice when using it.
Linux lets you have a choice. You can make your desktop look like pretty much anything you want, and can structure your windows and menus any way you like simply by changing the Window Manager currently in use.
Gnome and
KDE are both very user-friendly, MS Windows-resembling WM's. There are others, if you don't like the Start menu taking up all your screen real estate, aren't keen on the bubbly-windows look, etc, but those are the 2 biggies. Take a look around their websites, there is a lot of useful information there which can help you choose. Choices are a big part of what makes Linux fun
FTP:
Client isn't a problem - that's another usual linux built-in tool. As for an ftp server - there are a heck of a lot of them out there, and probably just as many threads on the subject here at LQ. Personally I'm fond of
ProFTPd, but Google and searching here is a good starting point to find one that looks good to you.
Dual-boot:
Not a problem. The distro isn't even what decides whether or not you can dual-boot, that comes from your boot manager (there are 2 major ones: lilo and grub). Completely personal choice decides which one you pick, although I'm fond of Lilo as I've found that its config file is very slightly more intuitive. Only one word of caution here - install Windows
first. Then install Linux. Linux boot managers don't mind booting windows also, but the Windows default boot manager will absolutely stubbornly refuse to boot linux, so you've got to be sure to install linux 2nd.
OS:
Well, I'm definitely more fond of Mandriva than Fedora for reasons which are arbitrary and would probably get me flamed if I listed them, but in reality they're both equally good get-to-know-linux distro's. Toss a coin if you can't decide
Installation:
Install Windows first. When you partition your hard drive, carve off enough for Windows to sit happily on and leave the rest unpartitioned space.
Once you're good with your windows installation, shut down and boot from the linux installation CD. Mandriva has a really nice installation process, very similar to how Windows does it, and you shouldn't have any problems. It should automatically detect the unpartitioned space, and ask if you'd like to install there. You can even have it automatically create its own partition structure, as Linux typically uses at least 2 or 3 separate partitions instead of Windows' single C:\ partition.
When it gets to the end and asks about a boot manager (Mandriva favors Lilo, if I remember correctly), say Yes, install it to the MBR; yes, there is a Windows installation here that I'd like to keep so please dual-boot ...and all things being equal you should be all set ^_^
When you reboot what should happen is you'll get a choice between Windows and Mandriva. Choose the one you want with the arrow keys.
Let me know if there's anything that I left out, was confusing on, etc...and welcome!