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The particle か pops up in various places (among others):

  * Turning interrogative pronouns into indefinite ones: なに -> なにか, だれ -> だれか, etc.;
  * Listing alternatives (like the English conjunction 'or');
  * Turning declarative sentences into interrogative ones.

These all seem vaguely connected to some idea of doubt or possibility. Is that
just an accident of having only so many short particles, or there some
historical connection between them? More specifically, was there some sort of
irrealis marker that links these usages?
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question
etymology particle-か
Common origin of uses of the か particle
~~I'm not sure what you mean by _"some sort of irrealis marker that links
these usages"_. If you could clarify that, I will update this answer post.~~ →
_(See the UPDATE below in response to the asker's clarifications.)_

That aside, the か marker is a general marker of indeterminate-ness (an
"indeterminer"? :) ).

  * After a statement, it makes that statement indeterminate: it indicates that the speaker is seeking confirmation -- it turns the statement into a question.
  * After an interrogative pronoun (like 何{なに}, 誰{だれ} etc.), it makes that interrogative even more indeterminate in a way -- compare the use of English _ever_ when used after interrogatives, such as _what_ → _whatever_ , _who_ → _whoever_ , and so forth.  
(Note: this is just as an illustration of the mechanics of turning an
interrogative into a non-specific noun. Strictly speaking, a closer
translation of _whatever_ into Japanese would be 何{なに}も, where the も maps
closer to the _ever_ part.)

  * When listing multiple possibilities or options, it again marks indeterminate-ness: it's A, **or** B, **or** C, etc., in an indeterminate fashion.

All of these uses are conceptually much closer in the Japanese. The idea of
these each being separate "words" that all just happen to be か is largely an
artifact of trying to translate these uses into English.

_Speculation:_

In classical Japanese, I am interested in the apparent semantic relationship
between か as an indeterminate / non-specific marker, and こ or こそ as a
determinate / specific marker. That said, I have not read any research into
this one way or the other: this is purely my own personal speculation.

_Addendum:_

Historically, the various indeterminate uses of か appear already in the oldest
Japanese writings, such as the
[万葉集{まんようしゅう}](https://en.wikipedia.org/wiki/Man%27y%C5%8Dsh%C5%AB) (completed
circa 759, with some sections possibly centuries older) and the
[古事記{こじき}](https://en.wikipedia.org/wiki/Kojiki) (circa 712). This has been a
feature of the language for basically the entirety of recorded history.

**UPDATE**

To clarify more about the irrealis or conjunctive uses, this _was_ more of a
feature of classical and Old Japanese. か as a 係{かかり}助詞{じょし} (binding particle)
could be used at the end of a sentence to:

  * follow a noun or a conjugable word in the 連体形{れんたいけい} (attributive form) to indicate a question, doubt, or a rhetorical question.
  * follow a conjugable word in the 已然形{いぜんけい} (realis form, used in modern Japanese in the _-eba_ construction that is analogous to a conjunctive) to indicate a rhetorical question.
  * follow a verb with the ぬ completion auxiliary to indicate a desired possible outcome.
  * After specific particles to indicate a rhetorical, such as のみか, どころか, ばかりか, ことか.

か could also be used within a sentence (not at the end) to:

  * follow a noun, tying to the conjugable word at the end of the sentence that must be in 連体形{れんたいけい}, to indicate doubt or a rhetorical question.
  * follow a conjugable word in the 已然形{いぜんけい} (optionally + ば), an adjective stem + み, a conjugable word in the 未然形{みぜんけい} (irrealis) + ば, or other conditional phrase, to indicate a question or doubt.

Note: I'm not personally familiar with these constructions -- this is a
translation of the content given in Shogakukan's 国語{こくご}大辞典{だいじてん}.
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answer