Running an Ubuntu variant, I presume the project is in /var/lib/Boinc/Projects
Can I just copy the whole folder, on a USB stick, and move it to a new pc?
Can I also run the project on both PCs if possible? (one will be faster than the other, so the first one will pass the results, while the other one will verify the results.

Possible?

You can move it to another PC, as long as that PC has the same operating system and as long as the PC you move it from doesn't work on the same tasks. That answers your second question, no you cannot have two of your computers run the same work and have them validate each other. Well, not unless you run your own project on just your own computers.

When you move the work, you have to move the whole data directory, including all sub-directories and applications within.

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And the server knows to which computer it sent the task.

And the server knows to which computer it sent the task.

That was my worry, if they did know, if I could still transfer the data or not..?

Alternatively, can I just plug my ssd in the new computer? I know the OS is pretty flexible on hardware changes.
I only worry about a changed wifi chipset and the fact that it's changing from a 4 core Intel to a 12 core AMD.

It's too long ago to remember, and BOINC is all different now.
Try it and see.

And the server knows to which computer it sent the task.

Which is why you move the whole data directory with everything in it. That includes the files that hold the deviceID.

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