(Sorry if this is a really long question, it said to be specific)
The company I work for has a number of sites, which have been running for some time with no problem
With IIS not respecting recycling settings: did restarting IIS/rebooting change the behavior?
I would advise using a non-tcp connection if you are still running the SQL instance on the local machine. SQL Server supports several protocols, tcp, named pipes, and shared memory are the more common.
Named Pipes
Data Source=np:computer\instance
Shared Memory
Data Source=lpc:computer\instance
Personally I prefer the Shared Memory. Remember you need to enable these protocols, and to avoid configuration mistakes I suggest you disable all you are not using.
see http://msdn.microsoft.com/en-us/library/ms187892.aspx
IIS Reset
In IIS7 there are two ways to configure the idle-timeout. Both begin by clicking on the "Application Pools" section and right-clicking the appropriate app domain. If you click the "Recycling..." option there is one setting. The other is in "Advanced Settings..." under the section for "Process Model" you will find "Idle Time-out (minutes)" which set to zero disables the process timeout. This later option is the one that works for us.
If I were you I'd solve this problem first as restarting the appdomain and/or worker process is always painful even if you don't have a 20 second lag.
Some ideas:
Just some initial thoughts on where I'd start to look based upon your problem description. I hope this helps. Good luck with things!