Watch your spelling of the word. There’s no fada over the first i.
I’ve never seen it used that way before but that doesn’t mean it’s wrong. I have never seen it used after “ag”. Usually it is “ag déanamh miongháire”.
Of the three sentences you gave, the first one is the one that strikes me most as being potentially authentic, but I am unaware if the “smiling” of the sun is a visual that exists in Irish. But the full answer is I don’t really know. As long as it’s grammatically correct, I guess there’s no real reason you can’t use it.
Is it correct to use ‘míongháire’ (smile) in the following forms:
The smile of the sun/ Míongháire na gréine.
The smiling sun/Míongháireacht na gréine.
The sun smiling / An ghrian ag míonghaire.
I don’t know if the sun could smile at all in Irish, i.e. if this trope exists in the language.
miongháire is a noun, it is not a verbal noun or verb, so: Déanann tú miongháire, tá tú ag déanamh miongháire. Or: Déanann tú meangadh gáire; Tagann fáthadh gháire ort. The word “miongháireacht” doesn’t exist – but if it would, it would not mean “smiling” in the sense of an attribute. Smiling as an attributive adjective is aoibhiúil, gealgháireach: an ghrian ghealgháireach, an ghrian le haoibh an gháire.
That’s correct. I think the OP was assuming that since “gáire” IS a verbal noun, that “miongáire” is as well, which isn’t a bad assumption but in this case I believe an incorrect one. And correct about “miongáireacht” as well. It is a word that could exist, but it would mean something like “smileyness, or the act of smiling”.