I'm having trouble with an app that runs on IIS 6, but not IIS 7.5 (as outlined in this question: IIS 7.5 Can't open Handler Mappings?)
I think it might solve the problem if I could run IIS 6 on my machine as well as 7.5. Can this be done? Will it cause any problems if I do this?
I think that you can not because both of them need to open and listen to port 80 and 443. So there is conflict for sure.
This is impossible. IIS 6 is an integrated part of Windows Server 2003 and IIS 7.5 is an integrated part of Windows Server 2008 R2. You can not 'install' any one of these, they are simply part of the package (you can only activate/deactivate them via the 'Turn Windows features on or off' dialogue).
However, what you can do is configure the 'Managed Pipeline Mode' of the application pool that is hosting your application. If you set this to 'Classic', it will revert to the IIS 6 pipeline mode.