The brake 'stiffening up' is simply because manifold vacuum (stored in the brake servo) has ebbed away. Nothing to worry about.
So, normally you would switch on, foot on brake pedal, start engine - Selespeed would shift to neutral. Or, you could switch on, foot on brake, and hold lever to the right, and that should switch to neutral.
If the brake switch was faulty, unplugged, or mis-set (perhaps by catching the pedal with a shoe on leaving the car), perhaps that would make the Selespeed think that your foot was not on the brake and therefore would not select neutral/allow the engine to start?
So it seems like the brake switch is a good first check. Check that it clicks when you expect it to, and that the brake lights come on (with the key on).
When the car has been sitting and you open the door, the pump runs for about seven seconds?
This is just a first thought, I'm no Selespeed expert.