For new dies (and any other reloading equipment) I use Hornady One Shot dry lube/gun cleaner. Spray it on, wipe it off, no soaking necessary. BTW, it's great for cleaning new reloading parts, but not such a good gun cleaner IMO.
As for your OAL, 1.12 should be good, I seat my 115gr fmjs to 1.125 (ideally anyway) and they work great. It depends somewhat on the chamber of the gun you're putting them through. If they're not chambering properly check the diameter of the case mouth of the loaded cartridge with a caliper. In my limited experience it should be .010-.015 LESS than the diameter of the base (.387) If it's not you may need to adjust your crimp. Or, you're belling the mouth of the case too much, the bottom edge of the bullet should just sit in the case.
Auto powder throwers are gonna fluctuate +/- .1 and that's just how it is, as far as I can tell. Unless you're loading for precision or loading on the ragged edge of maximum I wouldn't worry about it. And if you are, get a trickler.
All the above info concerns loading 9mm. As for lead bullets . . .
Oh, and WD-40 is not a degreaser, it's meant for water displacement and rust prevention, that's what "WD-40" means, Water Displacement, formula 40.