I've been freelancing off and on for about 5 years, and had a horrible time with this sort of thing early on. What I eventually learned was to manage expectations right at the outset. Establish that if you want to get paid for research, phone consults, whatever, that your client is on the clock. Often your first freelance clients won't have worked with a freelancer before, and may never again, so they are as unversed in this, probably more so, than you are.

With a recent client, I knew I was going to need an hour or so just to get caught up with their modifications to the platform (ecommerce ish) so I stated that up front, and actually asked for that hour no questions asked and then would provide a quote after I understood everything. They were more than happy to provide that and it gave me the room I needed to provide a very accurate quote for the whole project.

For your current situation? What I have done with a non-paying customer in the past is to walk away, but shoot a message regarding the work I'd completed, stating explicitly that I would not be licensing any of it to them without payment. To be honest? I had no idea how to even follow up on that, or if it had any legal weight whatsoever. I just made something up in my frustration, but in the end I got about 50% of what I had been promised and wrote a reasonable sounding granting of license to my work. Your mileage will obviously vary, but hopefully this will at least provide another idea.