Is it normal for a Derailleur pulley to "wobble" a little bit? Or does that mean it's time to replace I'm trying to repair my bike for the first time, so i'm learning.

It's a shimano deore set for a flat handle road bike. 

It won't stay in gear --- it keeps shifting while i ride, especially in the middle gears of the back cassette (first and last gear seem to stick better). It's especially an issue when i start pedaling backwards... the chain is completely undecided between 2 gears at almost all times.

I've already tried adjusting it the regular way, and it's not the issue.

At closer inspection, i'm noticing that the pulley wheel labeled "Shimano - Centeron G-Pulley" doesn't spin perfectly straight. It sort of "wobbles" while it spins, and so... my chain sort of wobbles to the left and right as it spins around it. This makes the chain keep jumping gears. I understand that it is supposed to be able to "slide" to the left and right a little bit (the "centeron" tech, lol)... but in this case... it wobbles ever so slightly.. .and i'm not sure if that is okay, and if it means i need a new pulley.

I took the part out and cleaned it. I put it back in, and it still wobbles.
Is my diagnosis good? Should i order a new pulley set?
The wobblings fine - its likely you need to replace the cable as it stretches over time. You might also have to reindex the gears - but try replacing the cable first