As we've said, the ONT and the router don't need to be side by side. There is nothing special about the ethernet cable BT provide, any standard cat 5e cable will do.
No, that's fine, I've got plenty of bits of that and I'm sure the man with the van will have a few too. They will actually end up being pretty close together eventually (the box and the router) because the place I want to ultimately put the router is actually quite close to where the current phone cable comes in (which I hope they can just more-or-less replace with the fibre optic line off the pole). But that I guess is another story.
In the meantime the router has to go a few metres away from the incomer point, probably a bit too low down for ideal working which is why I plan to move it later. The incomer point is at about first floor level which seems a decent enough location. The higher the better I've always been led to believe.
Could always just drill a small hole in the shelf to pass the cable through...