Past events indicate that whatever senior staff thinks is appropriate is what will be allowed or disallowed, regardless of what has gone before, what staff has said on Meta, or what written policy is. Stack Exchange is under no obligation to follow published policy in this area; if a policy doesn't produce the desired result for a particular situation, they are within their rights to change it to get the result they want.

So, there's no way to predict whether someone will be allowed to stand for election, even if there is explicit documentation one way or the other. As soon as someone who the senior staff wants to allow to run is in this situation, it will be allowed. As soon as someone who the senior staff doesn’t want to be elected is in this situation, it won’t be allowed.

It is appropriate that Stack Exchange controls who has power over the communities that fuel their livelihood. The only potentially problematic part is publicly humiliating someone who is not welcome to be a moderator because there’s no process in place to screen them out before they make public their desire to stand for election.

In the finest tradition of Dungeons & Dragons rules lawyers: in the absence of an explicit a priori policy against such a thing taking place, it must be allowed to happen.

Stack Exchange either needs to append the requirements for moderators, or accept that it may happen. Technically under this interpretation, users that were banned are allowed to be elected as moderator following the completion of their ban as well.

The OP asked:

Can a fired moderator be elected as a moderator?

• Should fired moderators be immediately eligible to stand for election, subject only to community approval of them returning?

Yes, when a firing has been proven to be without just cause

Just cause means a legally sufficient reason. Just cause is sometimes referred to as good cause, lawful cause or sufficient cause. A litigant must often prove to a court that just cause exists and therefore the requested action or ruling should be granted.

Just cause, in the employment context, refers to the employer's right to discipline or terminate employees for misconduct or negligence. In many states employers must at least show just cause for terminating you. For example, if an employer punished an employee without just cause, a Court can order the employer to compensate the worker. Just cause is legal jargon for a legitimate business reason, such as wrongdoing on the employee's part. Just cause is often a matter of interpretation by the courts or arbitrators.

In addition, the website: US.Legal.com, proceeds to list the steps an employer is expected to take before taking disciplinary action or terminating an employee's contract. It is, therefore, not unreasonable to assume a similar procedure be in place to safeguard the reputation and good name of a volunteer; especially one who was appointed by the company itself.

As was the case on January 30, 2019 when Monica Cellio was appointed (not elected) as a moderator on Meta.SE.

Monica is an experienced moderator across the Stack Exchange network and we thought it would be presumptuous to ask her to step up yet again; that was a little short-sighted of us because she's been rocking it here and has the cycles to come on board. If we make more changes, we'll write a separate post. Things have been going very well, so we don't anticipate that being any time soon.

From the same public announcement, [emphasis in bold mine]

These folks [the appointed moderators] have quite a bit of moderation experience, some since the dawn of time the concept of pro-tem moderation itself. All of these individuals have been leaders on their respective sites, as well as leaders in the broader Stack Exchange moderator community. They're fair, even-keeled and most importantly, they're incredibly great at disagreeing with Stack Exchange!

• This isn't a pro-tem appointment, this is an office that they'll hold until they don't want it anymore or lose the capacity to continue. While we're very deliberately seeking folks that can constructively disagree with us, 'constructively' is a very key modifier, and we need to see that over a period of time.

Consequently, it is safe to presume that the company's executives asked its Community Managers to perform a detailed background check on Monica Cellio before officially appointing her as a moderator on their website. In fact, we (the community) were later informed by a CM that Monica Cellio's record was unblemished until her moderating privileges and all her six diamonds were revoked on September 27, 2019.

The question we are asking today is:

Did Stack Exchange follow a procedure similar to the one below?

1. Did the company warn the worker in advance of taking action?
2. Is there a clearly communicated work rule which covers the conduct and which is reasonable and related to the orderly, efficient and safe operation of the employer’s business?
3. Did the employer investigate before taking action?
4. a) Was the investigation fair and objective? 4. b) Does the supervisor serve as prosecutor, judge and witness all rolled into one?
5. Is there substantial evidence that the worker is guilty?
6. a) Has the employer been fair and even-handed in its enforcement of the rule(s) in question? 6. b) Is there "disparate treatment?"
7. Was the degree of discipline related to the seriousness of the worker’s offense and worker’s prior work record?

(source: the same page at definitions.uslegal.com as above)

No one, except the executives and directors of Stack Exchange, can say whether they adhered faithfully to their own guidelines, while one would hope that the answer to questions 3 and 6 a) is yes, there is more than enough evidence to suggest that the answer to the remaining five is a resounding and clamorous “No”.

If, for whatever reason, the company directors and executives dismissed a moderator unfairly, the moderator should be immediately reinstated and be allowed to candidate themself for any moderator election they see fit. The company should not tarnish or slur the candidate's good name further by refusing the nomination in any future election, be it in one or twelve month's time.

The latest revision of the election rules and qualifications can be found here:

https://stackoverflow.com/election/11

Notably there are subjective requirements:

Our general criteria for moderators are as follows:

• patient and fair
• leads by example
• shows respect for their fellow community members in their actions and words
• open to some light but firm moderation to keep the community on track and resolve (hopefully) uncommon disputes and exceptions.

So a moderator nomination may be withdrawn if they have violated any of those.

In addition to reputation and badge requirements, there is one objective requirement that may apply to situations like this:

• …and cannot have been suspended during the past year.

So even without changing the rules, they merely need to suspend the moderator for a short period of time to prevent them being electable for one year.

I don't know if the act of firing a moderator counts as or results in a suspension, but they could add that, again, without changing the election rules.