First, let me say thank you for all the excellent comments so far! Let me explain what I was thinking tactically on some of the turns in question, and folks can tell me where my thinking went wrong.
Turn 3: I know that later I will want a Trade Route, and since I hit neither of my opening purchases on turn 3 I expected to have very few turns below $5 after turn 3. Hence the Trade Route. With Duke, Spice Merchant would have been a worse choice, and I didn't think another Cutpurse was necessary. The other option was Silver, but like I said, I was not expecting to be below $5 again for a while and wanted to get the Trade Route into my deck.
Turn 11: By this point, I know that my opponent is going for Provinces rather than Duchy/Duke. So, the loss of a Copper does not hurt him that much. Playing the Trade Route allows me to get the green and a Copper, mitigating the effect of the trashing. On reflection it probably was the wrong play, but that was my thinking.
Jester: I didn't think about Mine as an option; that may have been better. My thinking with Jester was that it's a Silver and it is almost guaranteed to give me cards I want, since even Coppers are good. I don't know how my deck would have held up if my opponent went Jester, though.