Can I use my IPAD without a sim card?

Signing out of his Apple ID will not entirely remove him from the iPad. All apps downloaded with his Apple ID will remain tied ot it and require it update always.

If you want to completely remove your son from the iPad and use it for yourself, follow the steps here and set it up again from scratch: link-> What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support

And, no, it won’t affect his iPhone at all.

Yes. The iPad does not require a SIM card to work. The SIM card just lets it connect to a cellular service provider's network for internet access but is not required.

You can do anything that requires an internet connection over Wi-Fi. If the videos have been dowloaded to the iPad already you don't even need Wi-Fi.

Yes unless you are connected to a Wi-Fi network. If you have a SIM card installed to your ipad. You don't need to remove it. Just turn off Mobile Data in Control Center.