verifying that hardware and firmware are working properly
What the poster is asking is NOT "how can I test systems coming off the production line" which is an honorable and valid activity.
What the poster IS asking is "how can I test that my design is correct" and the only true answer to that is
you can't.

Just remember how many posts we have seen here "this is right, it tested OK, why does it not work now?"

Oh, how many times have I seen
"you do not need a MAX232 (equivalent) I did not use one and it worked, now with my new PC it does not work, why"
"I have been overclocking my '51 for a long time and everything tested, with this new batch of chips it does not work, why"

And the list goes on of things that "tested OK" but proved not to work.

So, to answer the original question
What is a good method for ... verifying that hardware and firmware are ... properly?
I would say a peer review is the best I know if the peer consider it a valid excersize

Erik