Does Microsoft recommend deleting and cleaning up the Default Web Site instance and extra Application Pools from production web servers?
Depends on what the server is going to be used for.
Some application installers assume the presence of the Default Web Site and DefaultAppPool, and these installers won't work properly if you've deleted the site.
Some might not even work properly if you recreate the site later before running the installer (unless you recreate it with the id 1- this was harder under IIS 6 than it is in 7).
If the server's to be used for hosting many websites and you're unlikely to run an installer that requires the use of the Default Web Site (Exchange and SharePoint in past versions used to do this), safest option is probably still to Stop or otherwise disable it, rather than to remove it outright.
As of IIS 7, any new websites created through the GUI default to getting their own App Pool, so any apps running under that website will run in that App Pool and not DefaultAppPool.
DefaultAppPool is usually a secondary consideration, and won't be used except by apps that explicitly name it or rely on its presence.
Normally I just stop the default website if it isn't going to be used for anything.
Yes, anything under the Default Web Site is still on your disk and virtual folders can be remade on another website with Powershell commands if necessary, for instance when creating an Outlook Web Access front-end.
An application pool is a pool for applications, if you remove all applications it would serve no purpose and can be removed. You can create those at any point, and they would be the same as the default application pool upon creation; as they would be
I think not. Also you don't want to delete it, you can limit access to this site to a particular ip address.