"Am settled down" is OK colloquially, too, using "settle down" as a transitive verb and in a passive construction. I think that use isn't quite as established though.

"I will contact you once I settle down“ is correct. You could also say "I will contact you once I've/have settled down“. The rule is this: Time phrases such as these: Before After As soon as Once When or conditional markers such as these If Unless Whether Providing need to be followed by a present tense when they appear in future sentences. So you can either use a present simple - 'once I settle' - or a present perfect - 'once I've settled'. You can't use a past tense here. Your friend is wrong.

"Once I settle down" is good. You would only use "settled" in the past, and the rest of it would have to be in the past, too: "I said 'I _would_ contact you once I _settled_ down' ".