Your situation is one most designers/developers have gone through once (at least some kind of disagreement with a client). So don’t worry! This is a typical example of what’s so important about good communication between you and your client. Not just to get clear what your client wants, but let them be aware of where their wishes come from and what other/better solutions can solve the problems they are facing. Having the requirements clear and written down in a contract can prevent you from disagreement on the final product.
These days you can expect from your client that they want to see the thing that they invested in to show it (off) on their phone or tablet. It is possible that you have good arguments not to have it scalable for these devices. But just saying that it's stupid is not very convincing, better have it substantiated with results from experiences or research (your own or from renowned sources).
How to solve this conflict? This really depends on what's on paper and how well you can get along with your client. But respect the situation and your client. Try not to be the one who is right, but to work towards a solution.
How to prevent this in the future? I think you have a lot to learn in web design/development. When you say...
squishing 10 links into a smartphone screen is stupid
...you should learn about information architecture or responsive design. EDIT: Because of one of your comments I now have a good sense of what you want to know:
"what questions do I ask the client and how specific must the answers be?"
In your situation first thing I would like to know is if it is relevant to spent another x hours to make it mobile friendly. So ask your client who will use the site, why, how often, where etc. or do some research yourself. Try to get a picture of the audience and what the websites purpose is to them. If it's just an informative site, you might want to find out what the most important information is to find quickly on a mobile device. If your client will benefit from people finding this information while on their way, you have a good stance in advising her to invest a bit more in a mobile version.