Does iPhone Run off mains power?

Does the iPad have this issue too?

Impossible to provide a good answer without knowing how old the battery in your phone is.

A processor intensive app like a painting app will inherently use more battery power than surfing the web will. If the battery is not near the end of its natural life, there should be no issue, but there is never going to be an assurance of X minutes of app usage in any circumstance.

Put it another way:

If your battery is dead, it will not run even when connected to wall power until there's sufficient battery power.

It will charge while using it.

It's a no. It runs on battery but when connected to wall power it will charge at the same time.

You can use the iPad charger for your iPhone and it's double the power.