You'll get a lot of useful suggestions here, and sound like you're already working in the right directions. My contributions:
1. You're right on the money listening critically to your patches. That's where you want to fix bad tone, and you won't regret the effort. That's onstage or in the studio. Using PA eq on a bad guitar sound is like spraying Febreeze on a turd. It's still bad.
2. Definitely get your PA send and stage sends on different outputs. SO nice to have the flexibility to turn up or down without affecting the audience.
3. I haven't tried the monitor solution you have, but never got really bonded with my Axe II tone until I got a good one. A bad monitor forces you to make corrections to your patches that sound bad when amplified through a house PA that isn't suffering from the same limitations as your monitor.
4. When setting gains on your FOH board, use your loudest tones, played hard. Do this right and ragging your FOH inputs is a thing of the past.
5. Watch those drummers. lol They are NOTORIOUS for barely tapping the drums (especially the kick) in sound check, then POUNDING them in the gig. It's a drummer's way of making sure their kick and snare come through at play time. Tell him to quit hitting it like a sissy. Always works for me!
6. Score a decent sound man, even if only for an occasional gig. Once he has a kicking sound, make notes of the way the board is set. It will be invaluable when you have to "go in blind."
Most of all, don't give up. It's awesome gear, and more than capable of making you proud once you get the kinks unkinked!