How can I tell if I need a valid SIM card (active phone #) on my old Smartphone to use an App? I gave my old Moto RazrM phone to my daughter. The phone and Mobile internet had stopped working. So it only has wifi. It also has not valid phone "account".

Will this limit her ability to use Apps?
If so, how can I tell when that's causing problems?
(Several apps/games she's downloaded are not working. And at least one of them worked previously.
The phone service (which includes texting) is a feature of the Android.  Tablets don't have this feature.

Without being connected to a phone provider will cause you to loose features the service provide.  However other features of the Android OS will not be affected.  You don't need phone service to use many apps.  Also there are many apps that can be used without Internet service (as long as you have a means (such as USB tethering to get the app on the device.

In other words the Android device should function very similar to a computer that works offline with applications installed and online when connected to the Internet.