What you are looking for is
XAMPP Portable Lite
XAMPP is a "Apache distribution containing MySQL, PHP and Perl" and you can use it for developing your website on your local pc.
You need to create an A record for sqldb1234.db in your DNS area for GoDaddy.
Once the A record has propagated you should be able to use the subdomain sqldb1234.db.mydomain.com in your connection string.
If you're talking about dedicated servers, then it won't be such a big problem to transfer the database.
You can use SSH in order to transfer it directly from one server to another, without even downloading it. Then you can import it in a new database from the command line pretty fast.
If you don't want to worry about storage and performances, then go for ...
You are not allowed to do this. AS soon as your website is out in the public you will need to use subscribe to the SPLA from Microsoft which is the Service Provider's License Agreement. Basically, you will be paying monthly for the use of SQL server, and the Windows server as well if you are also using Windows to serve up your application. Look for ...
If these are your own websites and you have a dedicated server, you should be able to use SQL Server Express edition which is free and can be used in production - but there is a limit to the memory and CPU it can use (and some other features), but for basic websites it should be enough. Are there features that you need in the web/standard editions?
Since you mentioned you might be open to considering an entire new solution to the problem, my preferred approach here would be instead of connecting via ODBC from your linux server to the SQL server,
Setup Apache/IIS and PHP on the SQL server,
Create an API written in PHP to perform the database queries required via ODBC but executed directly on the SQL ...
What you need is any kind of a so-called "WAMP stack". There are many solutions out there to choose from, see a comparison of WAMPs.
Although the most popular one is probably XAMPP, my recommendation would be to use Uniform Server instead. Uniform Server is much lighter and easier to manage than XAMPP. And it doesn't need an installation, so you can even ...
Not sure how GoDaddy has their servers set up, but many/most web hosts run dedicated MySQL servers, offloading the DB workload from the web server. As for security, SQL Server 2008 at least supports cell-level encryption like Server 2005 as well as full database-level encryption. Windows also supports file-level encryption via EFS and BitLocker. Given these ...