What @thwe said.
Aligning two rails on the same plane (assuming you align the frame well), is much much easier.
Oddly one of the very interesting bits of design feedback I've heard, is put the rails under the extrusions, not on top. That way they don't collect dust.
Seems silly, but its actually really nice.

Not using coreXY would be more rigid. Don't get me wrong, H-bot (or a weird monstrosity I designed) would be less stiff.
Basically, with the type of bearings, mounts (I'm assuming metal) and frame you've selected, all of your lack of rigidity will be from belt stretch. Two ways to minimize that stretch are to reduce the length of belt and increase its width. CoreXY uses a longer belt than you would have on a conventional cartesian/flying X motor. The flying X motor adds more weight and wiring complexity, so it might outweigh any advantage...
TLDR:
Wider belts is the easiest way to get more stiffness.
Its also really important to design in a way to adjust the belt tension!

2
I wouldn't.
Having the bed be structural and heated is a big challenge. If you are going to heat that bed (which will take lots of power, look at an SSR) then you want to both mechanically and thermally isolate it from a separate frame that is the actual Z stage. I've had great results just using nylon foam as a structural spacer. It can be glued, is a great insulator and rigid enough to work well mechanically. Its still a foam though and will happily absorb any expansion of the bed.
Screws and rails in-axis, its theoretically better, I think your bigger issue is that you seem to be relying on the motors own bearings to carry all the vertical load. They are not designed to do that, you should think about including a thrust washer.

3
How much does that gantry weigh? I'm guessing 450g for the extrusion. Plus the weight of the rail, rotational mass of the motors etc, if you are worried about the weight of the pulleys, don't be. If you mean the extruder? Its more significant, but the difference is going to be small compared to the mass of everything else, at least for acceleration not involving X.
If you really want to get higher speeds use bigger motors. NEMA23's are not much more expensive and produce vastly more torque. I don't know why everyone is so fixated on NEMA17's...

4
How fast do you need to go? 24V, 32 tooth GT2 pulleys and NEMA23's get me up over 500mm/s... Which given the huge mass of my extruder, is quite scary.
More voltage is better for speed... And you can always just use the Duet3's heated bed circuit to drive your extruder and only supply 24V to it.
You shouldn't need to actively cool the steppers if they are big enough. People tend to have to when they undersize the motors, overdrive them to get enough torque, then discover they get crazy hot (although good steppers will run hot enough to burn you without being damaged themselves).
You could also get an external relay or SSR to drive an extruder heater, although I'd try and avoid that myself. There is also the Toolboard 1LC if you can get one.
I'd suggest trying 24V at first.

5
How are you actually connecting your uprights to the crossbars? I don't see anything for that. But there are lots of easy ways to do that. If you haven't already go look through Misumi's website and see the millions of options. Including getting them to machine stuff into the ends of the extrusions.
The rest seems fine, although I wouldn't use u-channel for the gantry. If you change it the way @thwe suggests you can use flat brackets, which is way easier.

And to echo what @whopping-pochard said, if you move (in fact you can probably just remove completely) the 'front' crossmember, you get a much better view of the printhead/nozzle.
Similar theme, with the extruder mounted on top of the gantry, you have very little space around it for things like fans for cooling prints. Or fans to cool the extruder itself.

Don't forget to plan for filament mounting, and electronics!