When a door is opened partly it seems correct to choose the word "ajar". If the door is widely open, which words are correct to use? For example, can "broad" be used?
No. If you look up a dictionary, it will tell you that broad means wide in extent or scope. This implies that it is usually used with abstract concepts such as a project or one's knowledge.
My best suggestion here is what bytebuster said: wide-open. There are also less common terms like agape (when used predicatively) or gaping (when used attributively), although gaping usually refers to the open space created by the wide-open door than the door itself. I do, however, suggest you to follow bytebuster's advice unless you are composing serious texts.
As others have said, The door was wide-open means that the door was completely open in a physical sense. However, when converted to My/his door is wide-open, the expression often becomes idiomatic, meaning "People should always feel free to talk to\approach me/him". (This idiom might only pertain to North America English.)
As well, I am quite sure that gaping always refers adjectivally to a gap or opening in something that is not (usually) meant to gape/be opened. For example, There was a gaping hole in his parachute -- there wasn't supposed to be a hole. Similarly, the verb gape is typically used to mean stare, both of which refer to a kind of looking at that isn't supposed to happen (i.e., I couldn't help gaping at Princess Diana as she walked along the street; for the verbs gape/stare, to do so is socially-unacceptable/impolite--but as Hillary Mantel has recently pointed out, irresistible.