stock rocker spacing.

With new rockers, you may end up with the same thing. As said, they are mass produced and that side-to-side tolerance is not controlled. I have measured that much with the hydraulic rockers. Use shims to tighten up the slop if you like, but these ran with some side-to-side slop. The shims will be a bit larger than .875" ID. I personally would not swap around used rockers on a used shaft as they tend to wear in together.

Check your shafts for excess wear where the rockers ride with a caliper. I would consider changing the shaft for more than .003-005" wear or thereabouts for an engine I cared about.

Up and down should be zero with everything setup AND the engine run in, or pre-oiled and turned over a few times so that the lifters are pumped up. With fresh lifters that have never been pumped up, you are going to find the piston movement inside the lifters gives an up and down movement on the rocker. You may also find that with any lifter that leaks down after being run. So the only way to check that IMO is to look at the pistons inside the lifters on each cylinder one at a time with that cylinder at TDC of the compression stroke (not at overlap) and look to see that the lifter piston is pushed down into the cylinder body from the retaining clip by a short distance, perhaps 1/16".