The stabilisers can be a bit tricky. Even the plate-mounted stabilisers protrude a bit under the plate and can interfere with components on the PCB.
With Costar-style stabilisers the wire has to be on the side of the LED cutout in the switch. The Costar stabilisers do not sit in holes centred on the center-line with the switch but slightly offset from it. That offset has to be towards the side where the LED goes.
With Cherry-style plate-mounted stabilisers, the wire is practically
on the PCB. So there can't be anything on the PCB around the switch and a little bit to the sides. The switch orientation does not matter, though.
BTW, that PCB of yours has quite a few components on it. Are you doing debouncing in hardware?
Most of us do it in firmware on the microcontroller: One simple way to do it is to scan the matrix every millisecond and wait for a signal that is stable for at least five milliseconds (five samples) - but you could use a longer debouncing time.
I don't want the keys to sit so high so my plan is to put a piece of metal that surrounds all the keys. As I understand, the standard 1U keycaps are 0.750", or 19.05mm square. How much space should I leave between the keycap and this other metal plate?
When I have built keyboard cases, I have used the rule of thumb: (19.05 - width of keycap) / 2.
With most keycaps being 18 mm wide, that yields about 1/2 mm of spacing around each group of keys.
That should provide the same spacing between border and keycap as between keycap and keycap.
However, I have made my keyboard cases by hand and adjusted things along the way...