Responder cue-bid and raise of doubler's suit: invitational; Responder cue-bid and bid a new suit, game forcing. Doubler cue-bid (minor) after responder's cue-bid, game forcing. usually 4-4 majors with extra strength (K or more than minimum dbl). Doubler cue-bid ♥ (after pd's cuebid) shows interest in minors or NT, usually 3 ♠ only. While dbler's partner's cue-bid should show game forcing strength (opening hand).
I would bid 3♠ only. With this shape and strength (and opp's bidding), I don't think we can make 4♠ (statistically). With my holding of ♥, I am not afraid of opp's to go to 4♥. Opps go to 5 of a minor? Fine, I'd rather defend.
I would bid 3♥ at my second turn. Reasonable ♥ suit, if we belong in NT, I am telling where my strength is and better played on his side. Assuming I bid 2NT for some reason, now I might bid 4♥ only, thinking of 5H though.
Basically agree with Ben. 2♣ is out (which is michael). Double bid NT (if pd responds 1♠ or even 2♠)? What if pd responds 1♦/2♦? This hand is not as good as a typical 18-count hand: half of the strength is opp's opening suit, AKQ only worth (at most) 3 tricks. Pass? a little too shy and what would you do if pd balanced with 1♠? 1NT or 2NT? That leave 1♥ as the only choice.