How many times did we see this happen?
 

How many times did we see the Technodrome actually move? I know we saw it once in the first ep, and with quite an entrance of blasting through a wall to make its first appearance.

It ask because it seemed to change location quite a few times (caverns->lava chambers->Dimension X island->Dimension X wasteland->underwater->the Arctic, not necessarily in that order), but I never saw true moving going on.

On a related topic, when they were buried in hardened lava on the Dimension X volcano, how did they ever recharge it and get it free? It seemed like they were always trying to do these two things but never quite could.

I do remember they used a satellite-based laser to get themselves out of the glacier in the arctic, which didn't end too well because it got stuck and they fell through the ice to the bottom of the arctic sea.

Let's see, it moved in the following episodes as far as I know:

Enter the Shredder
A Thing About Rats
Hot Rodding Teenagers From Dimension X
Return of the Technodrome
The Fifth Turtle
The Big Rip-Off
The Big Break-In
The Big Blow-Out
My Brother, The Bad Guy
Rock Around the Block
Attack of the Neutrinos
Shredder Triumphant!
Turtle Trek

You can make any corrections to this (additions/omissions).

Re: How many times did we see this happen?
 

You're thinking of the second episode (Enter the Shredder), not the first. All you see of the Technodrome in the first episode (Turtle Tracks) is the inside.