Have the car painted first. Then assemble it. Install the engine from underneith only after everything in the engine comp has been attached. If you do it that way, there won't be much that needs to be hooked up once the engine is installed. All you'll have to hook up is the throttle cable, heater hoses, one main wire harness connector and some vacuum lines.
You can work from inside the engine compartment while bolting things to the firewall and inner fenders. You shouldn't have to lean over the fenders much at all.
I'd be leary of a bodyshop that would suggest painting a car after it was partially assembled. I guess if the engine compartment is already painted and the outside of the car is the only thing not painted that you could do it that way. It's not how I'd do it though.