”Mi-e foarte cald” wasn't accepted. I guess it should have been?
Yes, it should have been accepted. It's actually more likely that a native speaker would use the short form.
Eu sunt foarte cald=mi e imi foarte cald? Are those have the same meaning in romanian? In my native language they are not.
Keeping it simple " eu sunt foarte cald " is accepted.
Is "sunt foarte cald" also correct?
"I am very warm " = "(eu) sunt foarte cald"
"I feel very warm " = "(mie) îmi e foarte cald"
Duo is very right = Duo is not right.