Family traveling from USA to Nepal

Your itinerary is definitely eligible for 24 hour visa free transit and you don't need any kind of visa and you also don't need any money, photos or similar. This is entirely free of charge scheme.

You will print out your flight tickets and have those printouts with you. Also, you will declare at check-in in US that you will travel to Kathmandu, Nepal, via China using 24 hour visa free transit. After a short inspection, you will be allowed to board the plane.

Upon your landing in Shanghai you will approach 24 hour visa free transit counter where you will apply for a temporary entry permit because your next flight will be a domestic one. You will be granted a permit because there is no reason not to. You are perfectly within all the rules, so you will be given a green light for the continuation of your journey. Please, have in mind the fact that you should ask at check-in in US whether your luggage will be tagged all the way to Kathmandu. If you are traveling on a single (through) tickets this will be a case almost for certain, but you should ask for this detail just for a case.

By the way, your application for 144 hour visa free transit will not have any effect on your eligibility for your journey. If asked, although the chances are slim, you will just tell that you have confused and that you politely ask that such an application is ignored.

In the end... Your 24 hour TWOV will not be denied. Simply speaking, there is no a single reason why would they do that. It is in their interest that as much as possible of international passengers use their airlines and airports.