Translation:Yes, of course.
"はい" is bit more formal and can be used in almost any situation. "ええ" is a more colloquial but still polite way of saying "yes". it carries a somewhat feminine connotation.
There are some situations in which "はい" cannot be replaced with "ええ". here are some situations
1.) When answering the phone. はい、木村です Hai, Kimura desu Yes, this is Kimura
2.) When answering your door. はい、どうぞ Hai, douzo Yes, please (come in)
3.) When you respond to someone's request. はい Yes, (I will do it)
うん is a colloquial way of saying "yes" and is closer to the English word "yeah". It is very casual, therefore used only among family members or close friends.
... So basically, it's the same sort of "ee" as when we English speakers say "eeeeh..."? In other words: a reluctant 'yes'. A 'maybe' without actually saying "maybe". Like if someone were to give me something they'd baked, only it was so disgusting that I gag, and the person then says, "You didn't like it, did you?" and I reply, "Eeeh..." because I really didn't like it, but neither do I want to hurt their feelings. Is that right? Am I close? : )
Do Japanese people say ええええ when they unsure about something and they need time to think about it?
I've heard "えと" a lot in animes. I searched on the net and it seems that they use both"えと" and "あの". Apparently, they both mean "umm" but they are used differently. "あの" is used if you know what you want to say but don't want to be so direct about it. "えと" is used when you're thinking of what to say next.
We already have plenty variations as well though. Filler sounds/words while you think of what to say...
"Um", "ah", "uh", "well", "like"...
Based on Drackard's comment, if you'd like to compare directly, then yes - あの would be like "Um" (trying to grab attention), and えと would be like "Uh" or any "thinking sound" (stalling for time).
I'd have said that "well" is used when you know what you want to say but don't want to be so direct about it. "Er" and "erm" are used when you're thinking of what to say next.
That is in British English, though, so it might vary depending on which sort of English you are speaking.
です is not an existential "to be", but rather a descriptive "to be"; i.e in cases where someone is something. If it were as you understood it, the translation would have to be "Yes, I am of course" - with the implication that an "of course" is a state/thing one can be.
With "of course it is", Duolingo means "it is [like that/as you said]".