Can I get the ipsw file off my iPhone 3gs and install it on my iPhone 4s? I still have my old iPhone 3gs (AT&amp;T) with iOS 6. I have a back up of it as well. I also have a iPhone 4s (Sprint) with iOS 7.

Can I grab the ipsw file off my iPhone 3gs and install it on my iPhone 4s? 

I will not install any ipsw from the net. It is not worth the risk of getting an infected or spyware file.
No, even if you obtained the IPSW, it would not install since there are different IPSWs for different devices.

You can check the authenticity of an IPSW per this question:


Can an .ipsw firmware restore image&#39;s origin be verified?