This behavior of mine is familiar? I don't understand why the μου is where it is.
This is an excellent question! This phrase, somehow, only has one meaning: μου goes in the "μου είναι οικεία" group of words, not with "η συμεριφορά". This could also be "είναι οικεία σε μένα" = "is familiar to me". Because Greek personal pronouns have weak and strong forms (see also my comment here), 'σε εμένα' becomes 'μου' but goes in the beginning of the phrase.
If I wanted to talk about my behaviour, it would be "Η συμπεριφορά μου μου είναι οικεία", but the whole things sounds messy because obviously everyone is familiar with their own behaviour! So you cannot omit the second μου, it's the one that makes the whole familiarity phrase work.
That doesn't make sense to me but I'll try and memorize it anyway. Also, when speaking this is there a pause after συμπεριφορά so that the μου is sort of separate ?
is there a pause after συμπεριφορά so that the μου is sort of separate ?
Maybe? I don't see how a native speaker would misinterpret this (sorry!) so I wouldn't say there needs to be a pause. You can say the whole thing at once and the meaning is still clear. A pause would be more obvious if the phrase was "Η συμπεριφορά αυτή μου είναι οικεία", but I think that's because emphasis is put on αυτή forcing you to pause after it.
Τhere would be a pause before είναι if the μου was a possessive though.
Maybe someone has a better explanation, all I can think of is that you can't use "είναι οικεία" without saying to whom this is familiar. That's why μου is not a possessive.
To my knowledge you could make it clearer if you wrote "μού" in those cases, where "μου" could mean "my" or "me"...
Is this always the correct syntax? Could I say, "Αυτή η εμπειρία μου/σου/του είναι σημαντική."? This experience is important to me/you/him.
Yes, for the short, weak, version. Alternatively, είναι σημαντική για εμένα/εσένα etc. And you'll hear it as για μένα, για σένα, γι'αυτόν/ήν/ό etc.
Unless greek grammar changed recently...
With this specific acceptable English translation, the greek sentence is grammatically incorrect (and, because of the nature of the Duolingo site, this can be confusing even for a native speaker). According to the greek grammar, afaik, it should be "μού" with an accent...
Rule: The weak forms of personal pronouns get an accent when their role is not very clear for the reader (as it happens in this case)...
"αυτή η συμπεριφορά μου είναι οικεία" (ex. "... to my friends") =
"this behavior of mine is familiar (to someone)" (my behavior...)
"αυτή η συμπεριφορά μού είναι οικεία" =
"this behavior (of someone) is familiar to me"
"το παιδί μου έδωσε..." =
"my child gave (something to someone)"
"το παιδί μού έδωσε..." =
"the child (mine or not mine) gave (something) to me "
While some people consider "attitude" (στάση), and "behavior" (συμπεριφορά) to be interchangeable, personally I think there is a difference. A behavior could change, according to situation, but an attitude not so much. These words have a different meaning in Greek as well, so I'd suggest to keep these ones seperate. ^.^
But in this particular sentence συμπεριφορά could mean a temporary behaviour but also a more permanent attitude.