This is very odd. I had an email notification of a comment by SrGI2aed on this thread, which was very interesting. That seemed to be his second try, but I didn't get a note of an earlier version that I'm aware of. Now, although the thread is showing as having six replies, only four are visible, with nothing from SrGI2aed, and the reply comment from AliceSchee1 in which she thanked him for his trouble also not here.
I'm copying the SrGI2aed comment below to see if it stays. If it turns out that he deliberately deleted it I will delete this post. (Paragraph breaks are editorial.)
Well, I gave a long reply to this but it is not showing up! I'll try again, although I'm slightly concerned that I will end up muddying the waters rather than clarifying. It is probably best not to get too focused on alienable/inalienable distinctions, but to focus on the genuinely inalienable, i.e. things that are either attached to you or form an integral part of you. There is a hard-and-fast rule here - the possessive pronoun (mo, do, etc.) must be used. Thus m' fhalt, mo chas, mo dhruim, mo cheann, mo thòn. Everyone has to use the possessive pronouns here - native speakers and learners alike.
Then we have family members. Mostly you have to use the possessive pronoun with family members - m' athair, mo mhàthair, mo bhean, mo mhac, mo sheanair, mo sheanmhair, mo bhràthair, mo phiuthar, m' antaidh, m' uncail, etc. The exceptions are my husband, in which case almost all speakers will use an duine agam, and my daughter, in which case the vast majority of speakers will use an nighean agam. Duine and nighean both have more general meanings as well as the specific meanings of husband and daughter, and that may be at the root of the difference - all I can say to explain is "usage dictates".
It makes more sense to learn the different usages for the different family members, rather than try to understand the differences in terms of inalienable/alienable. I think Duolingo is correct in strongly preferring an duine agam and an nighean agam, and I would not recommend using e.g. mo nighean unless you were an extremely confident speaker embedded in a native-speaking environment in which mo nighean was the community norm - even then, an nighean agam would be fine.
I mentioned on another page that I would normally use the possessive pronoun with pets (i.e. with animals that feel like part of the family), but not with working animals or livestock. But like so much in Gaelic, usage varies. It cannot be assumed that someone who uses mo chù is more attached to their dog or loves it more than someone who uses an cù agam. Here I'd advise learning what Duolingo recommends and sticking with that until you are very confident in Gaelic, but being aware that other speakers may use a different form.
Then there is everything else. For everything that is not genuinely inalienable (i.e. a body part), or is a member of the family other than husband or daughter, and possibly pets, an x agam/d is an alternative construction to mo/do - and - very importantly - a vastly more common alternative. When teaching Gaelic as a second language, a key pedagogical aim is to rewire learners' brains so that the "weird" construction (in English-language terms) becomes the "normal" in Gaelic. This is particularly the case with mo, etc., which English speakers tend use as the go-to translation for my, but which in the Gaelic world is far less common than an x agam. But that does not mean that mo is wrong in most cases - in the wild you will hear mo chòta, mo thaigh, mo leabaidh, mo chàr, etc. as well as the much more common (am I stressing this enough?!) an còta agam, an taigh agam, an leabaidh agam, an càr agam.
But at this stage of learning, it's important to focus on rewiring brains to the most common Gaelic structures, so a practical approach is to think that an x agam for things beyond genuinely inalienables and most family members is a "rule" that applies to learners, but that native or confident speakers get a bit of a free pass on. Caraid comes into the category of "everything else" for most people, so, as Morag spotted, usage varies. This means that usage might vary from region to region, family to family, speaker to speaker, context to context, mood to mood, sentence to sentence. There is little point in trying to read too much about degrees of closeness of friendship between mo charaid and an caraid agam when you encounter these phrases cold. For example, sentence rhythm rather than implied closeness might subconsciously dictate one or the other - to take Morag's na caraidean agam vs mo charaidean aig karate, the latter avoids an unwieldy double use of an aig construction.
I'm hoping that this post doesn't also disappear...
(Spoiler: it did.)
There are clearly some days when it's best to stay off the computer... Apologies, I was in the middle of trying to edit my previous post yesterday (the one that finally worked) when I was called upon to deal with a family eruption, my multi-tasking skills failed and I ended up deleting the post by mistake rather than posting the edited one...
Anyway, I was going to add that I had thought of some other things you'd expect mo/do for rather than agam/agad - thoughts, opinions, mind (e.g. mo bheachd, m' inntinn), names (d' ainm - although "what's your name? what are your names?" in Gaelic doesn't use the possessive, but the definite form - dè an t-ainm a th' ort? dè na h-ainmean a th' oirbh?) - things that aren't tangible but that are still an "inalienable" part of you. Endearments like m' eudail. Majesty and suchlike qualities - a Mòrachd a' Bhànrigh (HM the Queen - "a" without lenition is "her"). Set phrases like Mo chreach! Mo nàire! (for shame!) Mo bheannachd ort! (well done, I'm proud of you).
There are no doubt lots more, but the overall point is that, as with a lot of things in Gaelic, what once might have been stronger rules are now breaking down, and the move tends to be away from mo/do to agam/agad. Once you get to know people's speech habits, you'll work out whether their use of an caraid agam and mo charaid implies different degrees of closeness or not - but it's not really sensible to read too much into such usage when you encounter such usage cold on the page. In most cases, if you use agam/agad for tangible things other than body parts, most family members and set phrases you'll be fine. Just don't talk about an làmh agam or similar, unless you mean the spare one you keep in a drawer...
There was some sort of glitch yesterday that nixed entire threads. It could have been related to that.
I noticed in Can Seo that the phrase given for "take off your coat" was something like "thoir dheth do chòta". Would that just be personal usage, or is there a grammatical reason why the agad form wouldn't be preferable?
Now you mention it I'd probably say that too about a coat you were wearing (closeness?). But on the other hand I don't think I'd bat an eyelid if someone used agad. I might be more variable about referring to one hanging in a cupboard? Now I'm confusing myself thinking too hard about it.
I've asked this in another thread but I don't think I got an answer. I absolutely definitely heard a girl in a programme on BBC Alba say "na caraidean agam" referring to the other members of an athletics club she was in. (Actually it was a voice-over as the child was actually Catalan and not speaking in Gaelic, but someone chose that for the translation.)