Yeah.. that cap could be most of your problem. I'd be very hesitant to start blaming chimney height and/or inside diameter too much for your significant problem of getting to temp. Fixing either of those might improve things somewhat, but you've got a pretty solid installation (my lack of perfectionism may be showing) there already and a lot of nice brick work. Rebuidling that chimney is something I'd avoid until I was 100% sure of it being the only fix.
I think you're suffering from:
- A restrictive cap. - Keep it off!
- Not fully seasoned wood - hard to fix but you can burn hotter and minimize the creosote ooze
- It's not cold enough there yet - That will fix itself shortly as you move out of shoulder season into real winter
- You're still learning. - Let the fire get hotter before you close it down and then don't close it down as much.
Keep us posted on your progress, changes, and learnings.
Enjoy the nice wood heat.
Squirt some bleach from a spray container on that brickwork and then it might come off with a stiff bristle brush.