Well, I'm familiar with Drupal, which is like Joomla insofar as it being a PHP CMS application. While I prefer working in Windows environments as I do a lot of ASP.NET development, I have to do a lot of PHP work for my clients too.
So I looked into Drupal on Windows/IIS environments. One gotcha was the .htaccess rewrite rules needed for clean urls, which uses mod_rewrite available on Apache/Unix. I assume Joomla does something similar, so you don't end up with weird urls like mysite.com/node/1/etc, and rather you get mysite.com/my-groovy-product/
On Unix/Linux with Apache, you are set, however with Windows and IIS6, you have to use the free or paid version of ISAPI Rewrite Full or ISAPI Rewrite Free.
If the Windows web server is IIS7, it comes with rewrite module, you don't need anything extra, but you'll have to modify the rules in order to match the .htaccess rules from Drupal, or in your case Joomla, since it expects a Unix/Linux environment with Apache or similar Nix-based web server.
Here's a link to the rewrite module of IIS7
http://learn.iis.net/page.aspx/460/using-url-rewrite-module/
I got the link from this article on Clean URLS on IIS. It's for Drupal but it should apply to Joomla and any other web application/CMS that uses mod_rewrite.
http://groups.drupal.org/node/25404
Some other things are that you may have to enable some PHP extensions if Joomla uses something special (ie. cURL, SimpleXML, etc), but Jodohost can help you with that if you need the PHP dlls enabled.
Ultimately though, unless there is a strong need to run in Windows, I suggest running your PHP app, in this case Joomla, in a Unix/Linux set-up rather than windows. Less issues to solve. For instance, some PHP apps only like certain configuration like to only run in ISAPI or FastCGI, if on Windows.
In my case, I'm trying to setup an ecommerce store for an existing ASP.NET site here, and the client wants to use Ubercart, the e-commerce add-on for Drupal since he is familiar with it, so that's why I've been looking into this.
As for the MySQL/database migration, I would imagine that would be fairly standard procedure of backup of MySQL db and importing to a new MySQL server.
I'm sure the Joomla community has lots of people saying how they got Joomla on Windows, if you decide to go that route. I know I read someone here running OSCommerce successfully on there Windows account.