No, modern Irish is the key to the meaning, and no native speaker would understand "clann agus Éire". I would suggest feaimilí for "family".

Dioghluim Dána is Early Modern Irish, but I suspect you are only interested in Irish from the last century or so. Regardless, such uses of the term have never ceased to be acceptable.

Yes, native speakers would absolutely understand what was meant by "clann agus Éire" in the context of a tattoo such as this. As for whether a native speaker would be inclined to get a tattoo like this themselves, perhaps not, though I'd never say "never". I don't know if any of this matters, though, as we were asked to translate what is very clearly an English-language sentiment, based on the English-language concept of the nuclear family unit, into Irish.

Clann is one of a number of options we explained to OP, and we were very clear as to its meaning. If they still want to tattoo it on themselves in the knowledge that it's not a perfect translation for "family", it's not because they've been misled here. Nor have we given them a translation which is unintelligible to native Irish speakers, just one that such speakers may not think is particularly native sounding, primarily because it's not a common sentiment in Irish.

For the record, I think if you want to use a word which is a direct analogue for "family", then feaimilí is a fine option, though I wouldn't have suggested it myself because I think suggesting a translation which clearly utilises a borrowing from English is missing the point of the request for an Irish translation. I also suspect it's just as unlikely a native speaker would tattoo it on themselves as clann agus Éire. Nevertheless, if OP is happy with the béarlachas nature of it, then feaimilí is perfectly sufficient. So too, however, is clann so long as OP understands this would be understood as meaning "children/descendants and Ireland".