Why does my dial caliper read a hair off zero even after I zero it?

The zero wandering is a common issue with budget older dial calipers and it usually means the tension adjustment isn't holding the gears tight enough. When you turn that zero bezel you are essentially trying to set the internal spring tension to a specific point. If it drifts, the tension has either backed off or the internal gears have begun to wear out slightly. For a small scale unit like that it's probably not a massive mechanical failure yet but it's definitely not perfect. You can try gently turning the tension screw—if it has one on the back—to tighten the engagement, but don't overdo it because then you'll bind the mechanism completely and it will be unusable. As for the slop you are noticing when you move the jaws quickly that's the backlash in the worm gear mechanism. It's essentially a gap between the teeth that allows that free movement before the dial catches. Dirt is a big factor with calipers that sit around for decades so a deep cleaning with isopropyl alcohol and a very fine brush is the first step. Once it's clean, a light application of white lithium grease or even a clock oil on the rack can help reduce friction and the feeling of that slack. Just a tiny bit is all you need.

Regarding the zero drift and the slop, these are typical signs of wear on mass-market tools from the late 90s or early 2000s. They aren't built to be sub-zero accurate instruments, but for checking lumber widths and dowel sizes for jig work they can still be perfectly serviceable if you stabilize the readings and accept a margin of error. If the internal gears are severely chewed up, no amount of oil will fix the inherent slop, but cleaning and tension adjustments are usually the only things you can realistically do on this scale of tool. Don't get discouraged since you're already in the repair phase.

One thing I'd check before blaming the gearing is the jaw faces themselves. My old Fowler had a tiny nick on one measuring face from the previous owner, and it made the zero look off every time I closed the jaws, which is easy to confuse with drift. Hold the jaws shut up to a light and look for a gap or a shiny burr line, then stone it flat if there's one. Also, some needle overshoot when you snap the jaws open fast is just momentum in the pointer, not slop, so I'd only judge the mechanism by slow movements.

That needle swing when you open the jaws fast is actually normal behavior, not slop. The needle has mass and the gearing takes a second to catch up, so it lags then settles. Any dial caliper does it, even my Mitutoyo that's lived on my bench for the better part of 15 years. The trick is just to open the jaws slow when you take a real measurement, and re-zero before each session. Fast swings are only useful for a rough sanity check anyway.

Update: I checked the jaw faces like one person suggested, and there was a tiny mark on one end. Fixing that zero reading helped alot. However, I'm still dealing with the needle swinging a bit when I crank the bezel quickly. Now that the zero is steady, is that motion just typical inertia, or am I missing somethig simple in the internal gearing?