I'd cash out for 12. Several things have happened here which may not happen at other tables: 1. Partner preempted 2. The ♣J finesse was taken 3. We're in 3NT, not 5♦
I don't understand how these 4 card major systems are playable. You could miss a 6-3 spade fit to play a 4-3 heart fit if you raise directly, and you could miss a 5-3 heart fit to play a 6-1 spade fit if you don't.
I don't think this is such a great hand for notrump, but there really is no other good call available than 1NT, particularly if partner is the type to double on 24(25) hands.
I would bid 3♠ if I was confident partner would interpret 3M as you described. My "default" action is 3NT with this hand since esp with a random (even an expert) I wouldn't be confident about this.
Without commenting on the meanings of 3♥/3♠ at responder's third bid, I assign most/all of the blame to East for not bidding 4♠ over 4♥. This makes it clear that East is interested in slam, which was not clear at any of his previous calls.
I think Frances generally has it right. Declarer play can be learned from books or other materials like Bridge Master, but bidding and judgment can really only be mastered by experience. Having players better than you criticize your bidding (and taking them seriously) and thinking about past bad results is a good way to identify patterns in your mistakes so you can avoid them in the future.
If you play support doubles and your'e playing up the line style then you have to play 1♣-(p)-1♦-(1♠)-Dbl as showing four hearts. There are books that talk about support doubles and mention this auction. There's a pretty good argument for it showing four hearts + extra values in the OP auction too, although I've never seen this explicitly mentioned in writing. The OP hand is clearly at least a king better than whatever the requirements are for such a double. I would bid 2NT if undiscussed.
Because this is WvR you can pass and get probably 200 which will be a great score if game doesnt make and wont be a disaster if it does. If this was equal or unfavorable I would bid 5♣.