Before I give any opinion, I'll start by answering your core questions:
Yes, you can run dedicated game servers on Windows Server Web ed.
The differences between web and standard:
- Web only supports 2 gigs of ram. Standard in 64bit mode can support 32gigs (and more?).
- Standard comes with more things that are better suited to local server environments (eg: active directory). If you want LDAP controlled Exchange email, you'll need Standard. Most web server don't need these.
- Web (apparently) won't support full-on SQL server versions. Express should run though.
Opinion time.
Dedicated and virtual dedicated monetary overheads on Windows servers are a lot... To the degree where you're paying more for the software than the hardware costs, at least for the first year.
Renting the software (as part of a managed dedicated server or VPS) is initially a lot cheaper, but over the course of a couple of years, will cost you about the same and if you run it longer, it'll eventually cost you more.
Shared Windows hosts can be good. I've been with a company called Hostek (Florida-based) and they've bent over backwards to make hosting a fairly busy site (around 6000 uniques a day) very cheap for me. It can also be atrocious. I've had bad hosting companies too. Shop around.
About a year ago, I dropped Windows at home in favour of Linux. I'm not going to enumerate the many benefits and drawbacks; I'll just tell you that that's when I stopped doing .NET in favour of more open Frameworks. I'm not using Django (a Python-based web framework). While you might not like it (or other frameworks - eg Ruby on Rails), I plead that you do check out what's happening in the open-source world before you plonk for anything Windows related since you already have the infrastructure available for hosting Django/Rails/et al.
If you wanted your own Linux server, VPSs start from around $20pcm. As I said before, severely cheaper than Windows counterparts. I now use Linode to host everything new I make. Highly affordable and they'll easily run dedicated games like your current set-up does.
Mono isn't an option for you. Not yet anyway. It does go some length to help people migrate their applications but it's still pretty sketchy on the ASPNET front. And as a comment says on another answer: the controls you want to use are strictly Windows-only for the moment.
Linux will consume fewer baseline resources than Windows will. On an old server (Windows 2000, IIRC) I had to administer, the core of Windows would consume anywhere from 100-200 megs of RAM. My current Ubuntu server eats 40megs. I'm not sure how much RAM you have to play with on your server but if it's a lower amount, you're going to fit a lot more on a Linux host. (Remember that if you have more than 2gigs, you don't have the choice of the Web Server edition)
It's clear from this that I'm a complete Linux super-enthusiast, but I know my needs differ from yours. ASP.NET is a great platform but it costs a lot of money even if you're splitting it between friends. You could opt for Windows... Or you could go Linux, donate a bit to the projects you use and buy a new plasma or something shiny for the lady.
Minimum requirements for all versions of SQL 2008 are here.
Server 2008 Web is listed as a supported OS, so you should be fine.
EDIT: I should clarify that it's possible to run SQL 2008 on Web Edition of windows server, but I'm not sure of what the limitations of doing so are. Server Web has more limitations than Server Standard, but I'm not sure what they are - be it hardware related or whatnot.
Best Answer
No version of Windows Server 2008 OS (Web, Standard, Enterprise) comes with Exchange. They all have basic outgoing smtp capabilities but if your want imap, pop3, or webmail you need to look for a 3rd party product.
A popular free mail server is SmarterMail