I think you are confusing termination in an at will state with "termination-with-cause" which means that you cannot collect unemployment, and cannot be rehired by the same firm.(or maybe I'm wrong)
When someone asks if you've been fired by a previous company usually they are asking about the second. They will call the previous company and ask if you are eligible for rehire.
No, not confusing it. Your original message was WRT paper trail, which is not required in all cases.
To your message here, fired means something very specific. Though fired can be for cause or no cause. And the reason of cause matters for unemployment.
Laid off means something else.
So really, there are four categories from an unemployment standpoint:
1. Fired - policy violation
2. Fired - "incompetent" (note: in the eyes of the employer)
3. Fired - no cause
4. Laid off
Both 1 and 2 are bad for new job prospects. 3 is hit and miss from a job prospect perspective, but still generally negative. 4 has no impact.
For unemployment in my state, 2, 3, and 4 will all let you collect (and bills back to the company who terminated the working relationship) where #1 makes you unemployment ineligible.
Circling back to your original message about paper trail, #1 is the only one that companies essentially always keep (or should keep) the paper trail for in my state, because it is the only one that is needed to to defend the company in an unemployment hearing if it ever gets there. For the other categories, there may or may not be a paper trail, and it certainly isn't required.
Depends on who is asking, what they ask, and for what reason. Though it is generally avoided.
If it is a simple reference check (never minding why a terminating company would be listed as a ref...) then as little information as possible would be given; possibly as small as "so and so no longer works here and that is the limit of what we can disclose with them" in order to avoid a bad reference lawsuit.
If it is an employment verification firm, depending on how rigorous the verification, it may come up by direct questioning and this should be expected by all parties.
Some companies do have strict policies about not disclosing some/all details. I am aware of a few firms, that wether good or bad, have policies against providing any reference, similarly, to avoid any potential lawsuit.
Now, when it comes to unemployment, if an ex-employee files a claim, the gloves will very likely come off. If the company can avoid a claim being made against their account, that is the difference of a lot of money on a recurring basis. So in this sense, reason very much matters.
When someone asks if you've been fired by a previous company usually they are asking about the second. They will call the previous company and ask if you are eligible for rehire.