The tuner screws are there to stop it twisting. They are mainly held in place by the bolt-down screw on the top, so they only need to prevent sideways movement, and only really need enough purchase to stop themselves falling out. As you are starting from new, though, I'd be tempted to fill with glue and sawdust or a wooden cocktail stick and redrill a slightly smaller hole. The kit tuner screws are generally made form very weak alloy, so you don't want to go too small a hole. If you've got a 1.5mm drill, then I'd use that. Try fitting the screws without the tuners in place first, and don't drive them all the way in, so that if a screw head does break off, then you should still have some thread to grip with pliers. I'd recommend looking to see if you can get some better replacements to begin with as the cross-slots will round off very quickly in those kit screw heads.

You may need to enlarge the neck mounting holes in the body. The screws should just fit through the holes without binding at all, and you certainly shouldn't have to screw them through the holes (relative hole size varies from kit to kit so you may not have to do this with yours). This allows the screws to fully pull the neck down into the pocket. If the screw has to also screw through the body, then the screw may stop turning before the neck is fully pulled down.

I don't know what sort of finish you are thinking for the neck, but any acrylic, poly, nitro or TruOil-style coating applied to the sides of the neck will make it wider and fit more snugly in the pocket. You don't want it super-tight to begin with, otherwise you'll never get it in the pocket when it's got finish on!

If the neck is very slightly loose in the pocket, then you can help get the strings running true down the neck by pulling left or right on the neck as you tighten the mounting screws up. You can often get a degree or two of adjustment, which is normally enough to get you sorted.