For some time I was planing to disassembly this valve because it was slightly leaking. Your post with pictures was a trigger to do it. But I wanted to do it more gentle way you did.
Everything what is needed is wrench no 14 (metric), Allen hex key 6mm, something to hold the valve (vise, adjustable spanner, wrench 15).
Wrench 13 could be used to open other side of valve, but I don't thing it's needed.
So this is my set of tools, I have used 14, spanner and allen key.
First using 14 and spanner unscrew one side, to be on safe side valve could be warped with thin leather before mounting it in spanner or vise.
My spanner don't have any sharp edges and there was no need to do strong grip in my case so I skiped it.
then using Allen key open second part
and we are ready for cleanup, this is completely dismounted but only left part is required
I have cut silicone gasket (ID 6mm, OD 9mm) to add some additional tension and placed it under Teflon one from one side only.
It's made from silicon cupcake form. It's about 0.6mm thin and probably shouldn't be thicker.
After mounting everything back, valve still looks stunning.