Does releasing a plugin under the AGPL force people to open source their whole WordPress installation?

I did cast a close-vote, since this question is a duplicate of the indicated question. (A Theme and a Plugin are functionally interchangeable as extensions of WordPress; therefore, any licensing implications of one would apply equally to the other.) That said, I will offer some thoughts.

Caveats:

1. The official WordPress Foundation stance is that PHP code in WordPress extensions is inherently derivative, and therefore must be distributed under GPL.
2. I do not personally agree with the official stance, and have documented why I do not believe the official stance has any legal precedent.
3. But I will not debate the official policy, or the matter of license derivation. While I do not believe licensing debates bear anything beneficial within the community, I state my bias, merely as a matter of full disclosure.
4. I choose to release all my WordPress-related code under GPL, for many reasons. Whether or not you agree with the official stance, if you want to be welcomed within the WordPress community, you should likewise release your WordPress-related code under GPL.

Now, on to your questions:

Would this force people to make public the source code of other unrelated parts of their WordPress installations? (I’ve tried to read the AGPL license but I’m not sure.)

As far as I know, GPL and AGPL are written explicitly to coexist peacefully. One does not “infect” the other, and clear boundaries exist between works conveyed in a combination/compilation.

Let’s just use some common-sense tests:

1. If your work is a derivative of another work, for which you are not the copyright holder, and which requires that any derivative works be distributed under a specific license, do you have a legal right to distribute that work under a different license?
2. Do you have a legal right to dictate the terms under which a work for which you are not the copyright holder is distributed?

I believe the answer to both of these questions is a resounding “no”.

Let’s set aside question #1, as it probably requires no further discussion.

Regarding question #2: let’s say a user has both your Plugin and the Akismet Plugin installed in their WordPress installation. In what conceivable way is Akismet a derivative work of your Plugin?

Let’s take it a step farther: in what conceivable way is WordPress itself a derivative work of your Plugin?

Would it have any legal effects if I publicly stated that I, the copyright holder, don’t consider other plugins and themes and WordPress being part of a combined work?

No. The only thing that matters is the actual text of the license under which you distribute your work. How you personally define “derivative work” doesn’t matter; all that matters is how copyright law and precedent case law define “derivative work”.

Note that, IMHO, GNU fails in this regard also. With the GPL, they define “derivative work” in at least three different ways – but regardless of how GNU defines “derivative work”, all that really matters is how copyright law defines “derivative work”.