Can I install any iOS version I wish by picking and choosing which firmware to restore? I'm trying to downgrade my iPhone from the current iOS to an older one, as per this question.

But every time I try, I always receive this error:


  The iPhone could not be restored. This device isn't eligible for the requested build.




I have tried many times with and without DFU mode. Any ideas regarding this?
Downgrading or upgrading to a no longer current version of iOS is generally not possible once Apple stops signing the old software, as is needed when iTunes is preparing to install or restore an iOS. This signing window typically is closed within days of a new version shipping. 

Putting a newer iOS on older hardware that never even supported that OS is near impossible.

As part of the iOS restore process, iTunes has to check with Apple's servers to 'sign' each install of iOS just like App Store apps must be signed to run. Once a new version comes out, Apple stops signing previous releases to keep everyone running the same iOS version and to prevent you from jailbraking your device again after restoring by forcing you to do an update.

If you had previously Jailbroken your device on iOS 6, it may be possible, providing you have saved your 'SHSH blobs' which can 'sign' the install instead of Apple. See this article for some general information around this topic.

This wikipedia page contains a list of iOS devices &amp; the latest iOS they can support - https://en.wikipedia.org/wiki/List_of_iOS_devices