Well you really need a bearing puller but there is a great video on this web site oh how to use a vise to remove an armature bearing. Hope that helps.
Tinker

Okay, I popped off the housing cover and it looks like all the gears are just greased.

I believe the problem may be an old & tired selector fork spring. I can pull the selector knob out of the housing quite easily and the the metal tip of the knob must get dislodged as the spring flexes around.

Aside from shifting between different modes, does this spring also function to keep the selector knob against the housing?

it is a grease in there. when you pulled the selector switch you should be able to see a ring that has a groove in it. you should also be able to move it forward and backward. a lot of times the selector switch is just worn.

if it does not move you will need to disassemble and check to see why it is not moving.

It's moving freely, and the grease looks okay. That groove you mentioned on the selector fork is formed by two clasps of a spring that attaches to the fork. It may be slightly different on Bosch's smaller rotary hammers, but that's the setup on this particular model.

The other related issue is that the knob pulls out way to easily. I know that on smaller models such as the 11222EVS/11236EVS, the knob locks into place, but I don't think that is the case with this model. Would a new spring remedy this issue? There doesn't seem to be an issue with the knob itself or the plastic hood onto which it mounts.