Going up and down, round and round and not able to move forward from this area (where 4 "gears" need to be deactivated) == if there's a step-by-step video or text guide, would really appreciate it -- it's baffling me for days..

My walkthrough with screenshots is here: http://tombraiders.net/stella/walks/TR9walk/details/research-base-4.html

I can't recommend a specific video, but there are several on YouTube: https://www.youtube.com/results?search_query=tomb+raider+research+base+elevator+puzzle

Good luck!

Stellalune -- thanks, again for those guides and videos -- finally got out of that zone (was quite claustrophobic!) -- but now, I'm in what seems to be the "slipperiest slope" of the whole game!  Is anyone else having an extremely frustrating time using a PC, mouse and keyboard (Windows 8) in the Chasm shrine trying to leap from the rope lantern to the climbing wall to get to the other side!?? Every time (and it's been thousands!) I attempt this using the arrows on the keyboard and space (jump) key, she swings to nearly the wall and falls! I hope there is a "save" file to enable skipping over this part, because it seems to be a technical (mouse and keyboard) glitch!?  Thanks, again for helping me reach this level, but now I'm tearing too much hair out!

TBH, I've only ever played the game with a controller, but I do have PC save files you can download if you want to skip that section.