I wouldn't beat yourself up just yet. You're keeping pace technologically with HC, which is commendable. Naturally, your methods could use a little work (like revolting to Caste/Pacifism earlier, and building either the Parthenon or the Pyramids) but I think you're doing okay. (I should mention that I'm pretty weak at the SE myself -- every time I've run a Specialist economy it's been completely by accident...and I won't even nitpick the decision to run OR w/o a state religion, I've done that myself as well!!!)I came to a simple conclusion.
I suck at the Specialist Economy.
On the other hand...gifting Engineering to GK...WTH were you thinking?????? Alright, alright, he can't build pikes w/o Iron, but you better make damn sure HC isn't trading Iron to him...
It also would've been a good idea to send your free missionary to GK's capital -- even though he's discovered Christianity since then, it's my experience that an AI likes to stick with whatever religion he's got in his capital. He probably would've converted eventually, but by that time his land will be your land.

I think it's a foregone conclusion that you need to attack GK ASAP, for all the reasons already stated. I would also consider trading Education for Drama/Banking or Banking/Compass (but don't trade if you can't get more than two techs) since HC will almost certainly go for Economics instead of Liberalism -- but this means you'll have to research Liberalism immediately. (Trading Edu to Genghis for Music or Compass would also be a good idea...just don't give him Guilds, for heaven's sake!!!!!)
Good for you! I believe it's imperative for everyone to stick with the difficulty they're most comfortable with, forget all this bullplop about "My Level Is Higher Than Your Level Nyah Nyah!!" That's why I stick with Prince -- I've tried Monarch, but it's too much of a struggle to remain competitive. I prefer to have a dominant leading position in all my games, but at the same time I like when it's a challenge to maintain that dominant lead. And Prince suits that just fine.Maybe it means I'll never graduate beyond Monarch level with Civ IV, but you know what? I'm finding Monarch is my level.
