Is there a method (or add-on module) for copying layouts between files and for comparing two separate files?

I will be rolling out a solution to 6 different companies, but I want to keep all the solutions somewhat similar.

You can copy layouts between files using a regular copy of FileMaker Pro. I guess to be more precise, I should say you can copy layout elements: you first have to create a layout in the destination file and (I'd recommend) creating all the layout parts (header, body, etc.) to be exactly the same size as the ones in your source file. Then pasting all the layout objects in to the new layout works pretty well providing your table occurrence and field names are the same in both files.

As for comparing files and checking for changes, there are three utilities out there to look at:

Inspector: http://fmnexus.com/inspector

FMDiff: http://fmdiff.com/

and BaseElements: http://www.goya.com.au/baseelements

Personally, I use Inspector and recommend it highly.