I do not think that I am completely clear on what you are asking, but I will give this a try.
Adobe allows you to use a purchased serial number on only two of your computers.
If Premiere Elements 10 on the two PCs represents the same purchased serial number, you are at your limit. And, to install that purchased serial number on a Mac (3rd computer), you would have to
a. Deactivate the program on one of the PCs
b. Buy a new serial number (either for Premiere Elements 12 or earlier version if you can find one online, maybe Premiere Elements 10).
Whatever operating system, the guideline is that you can edit a project.prel from one version in a later version, but you cannot then take the project.prel edited in a later version and edit it again in the earlier version. There is one rare exception, but it is not worth getting into.
Also, Premiere Elements Mac (any version) project.prel files are not compatible with Premiere Elements Windows (any version) and vice versa.
So, you are going to have to evaluate your situation and goals to determine what will work for you.
This is helpful.
So getting to the most basic question ... Anything done on a Mac isn't compatible with the PC? Is that correct? If so then I'll stay with 10 on the PC's and probably stay with Final Cut Pro on the Mac. I was hoping to add Premiere Elements to the Mac for home-office editing, but if I can't bring it back and edit it on the PC (and visa versa) it sort of defeats the purpose.
The project files (project.prel) from Premiere Elements Windows and Mac are just not compatible. However export can be selected so that the export is compatible on either. But, it sounds like you want the freedom to edit the project.prel (project file).
I am strictly an Elements Windows user. Never used a Mac computer. But, that is the party line from all fronts from what I have read here and elsewhere.
Wishing you success with whatever is your video editing decision.