What is the best way to punctuate a list of questions in a declarative sentence?

There is a rule that says if a question appears in direct quotation and this immediately ends in a sentence, the question mark should be preserved and the period omitted.

The way you have it is good. But if you're supposed to be following some house style, then read the manual.

Here's how I would write it:

This poses questions such as “How should I punctuate it,” “Are the quotes necessary,” “Are the commas in the correct place,” and “Should I have used a colon, or a semi-colon?”

I would say that the more important punctuation mark here is the comma, and you can't have both. Since the questions are obviously questions even without a question mark, and since you are referring to the questions as objects rather than invoking them as queries, it's OK to lose the question marks.

For example, you could imagine

The interview consisted of the usual "Where did you go to school" kind of question.

There is no need for a question mark here because you are using "Where did you go to school" as the name of a question to which you are referring, rather than as the question itself.

To some extent, though, your choice here is going to depend on what tone and cadence you want the reader to imagine in their head. Putting in the question marks will cause the reader to pause and raise their inner voices as if reading a question, which will have the effect of putting more emphasis on the specific question. Leaving out the question marks will cause the reader to rush through the list without pausing or imagining a question, which will have the effect of de-emphasizing the questions. So it's up to you what kind of melody you want the prose to have.

Quotation marks are not required in lists of questions within a sentence unless they are a direct quotation. Only rhetorical, self posed, or internal questions require a question mark. For example:

She thought, Shall I go to work? To the mall? Home?