Haigh, a Chairde!

I was wondering if anybody knows if starting an introductory title with an interrogative is appropriate in Irish. For example, an essay titled “What I Did Last Summer”…would “Cad é a Rinne Mé an Samhradh Seo Caite” be an appropriate title? Or “What I Listen to on the Radio” (Cad é a Éistim leis ar an Raidió) Thank you!

I've certainly seen Irish interrogatives being used as determiners, as they would be in English titles, but it always strikes me as quite béarlachas. That is to say, I would expect to see it in titles written by L2 speakers more so than in title4s written by native speakers. It may be catching on even with native speakers, I'm not in a position to say, though perhaps this doesn't matter to you. If not, then yes it's fine. I tend to lean towards constructions like "an rud/gníomh a rinne mé an Samhradh seo caite", though this is just personal preference based on what I believe I've heard more often.

That makes sense. Thank you so much for your input!

Cad in this position is wrong, it is "Béarlachas", except you really use a question with a question mark: "Cad a rinne mé?"
English what can be used outside (direct/indirect) questions, Irish cad cannot.

EDIT: I changed my mind, see below

I don't know why this is deemed to be Béarlachas. Peadar Ua Laoghaire has a chapter title in one his books "Cad a Tháinig as an nGorta" (in Sgéalaidheachta as an mBíobla Naomhtha, in reference to the plagues in Ancient Egypt), and the heading to Ch4 of Don Cíochóté is "Cad a Thárla Dhó Annsan". The heading to Ch3 is "Conus a Fuair Sé a Ghairm Ridireachta", and the heading to Ch6 is "Conus do Mheall Sé Sanchó Leis".

Phrases with cad are noun phrases:

Cuímhnigh in am ar [cad a thiocfaidh as].

Also phrases with cad can be direct objects, in the manner of the French ce que:

Do lég sé cad a bhí sgríofa. (Cnósach Focal ó Bhaile Bhúirne, 46)

Cad é a Éistim leis ar an Raidió - this should be an indirect relative because of leis: cad leis go mbím ag éisteacht ar an radio. or maybe: cad iad na rudaí go mbím ag éisteacht leó ar an radio. Cad iad na rudaí is gnáth liom bheith ag éisteacht leó ar an radio.

You have to remember that the British Isles form a Sprachbund: there are a lot of Irish phrases that are phrased like English, and not all of it is recent Béarlachas. Some of it is longer-standing and accepted Béarlachas, but some of it is just a long-standing similarity of expression. See https://corkirish.wordpress.com/idioms- ... ish-idiom/ for a long list.

I'm used to seeing mar ... in these contexts, e.g. mar a fuair Peig a scéalta, but there might be a slight difference in meaning.

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I'm an intermediate speaker of the Corca Dhuibhne dialect of Irish and also have knowledge on the old spelling
Soir gaċ síar, fé ḋeireaḋ thíar

Could it be possible that one of Ua Laoghaire's editors supplied the chapter headings? I would need to look at the manuscript.

Well, we can consider headlines as "Cad a Thárla Dhó Annsan" as question-like (title as question, the text of the chapter as the answer).
They are probably truncated indirect questions (short for: "Deirim leat cad a thárla dhó annsan" or "Fiafrófá cad a thárla dhó, deirimse leat") and so without a question mark (and without an interrogative intonation).

In this sense, Rosies title phrases are okay, too.

But outside of direct/indirect questions cad should not be used.