@nathalie-kranich yes, test illustration requests are common, and in my opinion, a good idea for both parties, as you can get an idea of what it will be like to work together before committing to a bigger project. Often, if a client doesn't request a test illustration, I suggest that we do one before either party commits to the main job.
Some thoughts on your questions:
Would I ever do a test illustration for free? Not anymore. I used to, and of all those times a client asked for a free test illustration, I never got the job. Plus, you're getting a client used to getting things for free from you right off the bat. In my experience, that can lead to your time not being respected or appreciated. So now, even though I'm happy and willing to do test illustrations, they are always paid.
How much time to spend on a test illustration? It all depends on what the client/publisher is willing to pay for a test illustration. If their budget is $100, I give them $100 worth of work ($100 divided by my hourly rate = the hours I'll spend). Of course, I plan accordingly and communicate clearly to the client what that $100 of test illustration will buy them. If it's only enough to do a sketch, not a completed illustration, I'll let them know so there are no surprises. If they want to see a full-color test illustration and their budget is $100, I'll let them know that that will get them a spot illustration, not a full page with background. Being up front and communicating clearly prevents a lot of potential problems!
So ... a lot of clients will ask for revisions in a test illustration, even if they're completely happy with the work. Why? This is a test run, so they want to see how an illustrator responds to revision requests. Is the illustrator going to be sensitive? Are they going to refuse? Are they going to be polite and accommodating? When I'm working out terms for the test illustration with a potential client/publisher, I clearly state how many revisions are included in that test illustration price. That lets the client know that I'm open to revisions. It also prevents scope creep or unlimited revision requests, which is a time suck.
To sum up: always charge for a test illustration. A test illustration will take a block of your working time, and that should be fairly compensated. It's also less risk for both parties, as they don't have to commit to a large project. Clearly state terms, price, and what the client will get. Also state ownership terms -- will you remain the owner of the test illustration? Will you retain your rights to the illustration or will the rights be transferred to the client? (I'd always retain rights, since they're only using this illustration to determine if they're going to hire you; it isn't going to be published so there is no reason why the client would need any rights to the illustration.)
Hope this helps and hope that this is a positive experience for you!