I intend to create content that answers questions, relevant to my target audience, in the form of articles that cover such questions. But how do I find out ..

a) what questions are of interest to my target audience?
b) how - meaning with which concrete question queries - it searches for them?

Example 1:

Assumed, I would like to find out what questions my target audience asks around ingredients, for instance "with which meals can I combine basil?" or "to which dishes does red pepper fit?"

Example 2:

Or assume, I would like to find out questions around fashion people are searching for, for example, "how to combine white jeans?"

So far, I have these ideas:

I could use the autocomplete / suggestions function of Google. But when I (to stick to the example) enter the keywords "basil" or "white jeans" there, it displays not a single question, containing the given keyword.

Do (ideally free) SEO tools exist that can find me question queries that contain keywords I provide?

Or maybe you have other approaches, how I could find corresponding question queries around given keywords I provide?

I let this question stew for a while since everytime I read it I thought "hey, they have their target audience so they know what that audience is here for."

So it's not like you don't have your target audience where you need the questions with answers to find the demographics.

Finally your post heads off the rails to me about SEO. The thing about such a site is you have your audience already so they are here for what you offered in the first place. Unless your post contained a blatent error about that. So for your target audience you create content for them.

Go to Keyword Magic tool and enter a keyword. On the right you’ll notice the ‘Questions’ button. Click on it and check the list of questions that contain your seed keyword.
The ‘Questions’ report works as a filter and takes into account all your other settings, such as Broad and Exact matches.

At the moment we identify questions for the following languages:

Once you have a list of question-based keywords, don’t forget to send them to Keyword Analyzer for deeper research!