I like this game/question because it's a very clean example of a fundamental dominion concept.
Playing engines like these correctly is all about finding the right balance between power and reliability.
You play very nicely until turn 9. 2nd Junk Dealer is absolutely correct.
After that unfortunately your decisions are really bad. DG already mentioned the buys I think are best on turn 9 (peddler peddler gold) and my alternative would be 3x peddler.
Why do I disagree so much with what you do?
Well, this is the point where you start drawing your deck. But you still have 4 coppers left, which will be eaten by your junk dealers. Also you have 2 schemes, so you don't have to worry about not drawing the cards in the right order. The result of this is that your reliability is now almost 100%. It's most certainly time to add power.
A 3rd peddler over a gold is a serious consideration. For you gold is clearly better - you draw it anyway. But your opponent is not set up to buy golds at all, and very much so to buy peddlers.
On turn 10 the right play would be to trash the catacombs mid-turn for a silver and buy two golds. You need power that badly. I guess that's the same point as turn 9, but the way you play it now 4(!) draws go to waste, and in your buy phase you waste another $2. Way too much reliability, balance is lost here.
Turn 11 you have $10 and buy village+gold. Silver+gold is better, same principle again. You really don't need more draw or buys!
Turn 12 I agree with double gold for $12.
Turn 13 - I think farmland gold->province, buy province is better but I'm not sure here.
Turn 14 - If you buy two farmlands and a moat (village -> farmland, gold -> province) you have 24 points to his 22.
And most importantly: the farmlands are all gone. His $12 is not so impressive without the option to farmland anything. I think you'll still win.
Turn 15 - I don't think you can win here, almost regardless of what you do / draw.