The clicking is the result of the Axis bar defeating the blade tang's "hook" closure detent. As it defeats it, the Axis bar, under tension, slams upward against the top inside of the Axis channel cutouts of the liners. All of my Axis locking folders do this. Some of mine, not all, do the same when closing, only this time the Axis bar slams against the bottom inside of the cutouts, creating the same click. No worries here, just how it does, and some more pronounced than others.
As for the wiggle in your Axis bar, mine tend not to have that much play. It's almost as if your Axis bar is too long for whatever reason. Not much you can do there (needs replacement with a shorter one), since the only adjustment other than replacing the Axis bar with a shorter one is putting in wider washers, stop pin, and spacers, which simply isn't feasible with most models that have a full or integral backspacer like the Grip family. I don't think it's a spring issue, I think it's the Axis bar being too long. One thing that can be done to compensate somewhat for this, short of having it replaced, is bending your Omega springs outward just a little to create more tension. This can be done by removing them, measuring the stock distance between ends, and bending outward to make them about 1/8 of an inch wider. Do this methodically, re-measuring as you go, and be sure to achieve the same measurements with both springs. Increase tension on the springs will make the Axis bar rock less noticeable, and increase overall solidity of lockup, though will of course make lock disengagement a bit more strenuous and may decrease Axis life to only 200 years instead of 300 years.
As for up and down play, several things attribute to this, some of which are self-correcting with time and wear. I've noticed over several models, that as long as there is very little to no pivot slop (tolerances between the pivot pin and blade pivot hole are reasonably tight), up and down play tends to decrease and even disappear the more the action is worked, and coatings begin to micro-wear and metals begin to mesh and deburr themselves. A smoothly and correctly-operating Axis mech actually acts to absorb any minimal blade pivot slop similar to how a framelock does, by creating a jamming of parts against one another at certain strategic contact points.
I personally prefer a Sebenza-like, hydraulic pivot tension on my Axis locks, and thus will crank on down on the pivot screw to full tightness after lubing and before the Loc-Tite cures. This eliminates side to side play, and any vertical play almost always takes care of itself with break-in. Just my .02, having owned many BM's over the years and being an Axis nerd - my dx is the Axis bar being too long.
Prof.