And am I allowed to use custom logo/text thanking them (at the title and credits) or just allowed to use the movie file they have?
There’s info on how you can use logos and such here: Unreal Engine Branding Guidelines and Trademark Usage - Unreal Engine
Can I refer to Unreal Engine 4 as Powered by/Made with…
“Epic Game Engine” or “Epic Game’s Engine” with the Epic Game’s logo instead?
Everything you need to know about how to use branding related to Epic/Unreal Engine is there