Regarding stations on the cable: space elevators are not static structures. The lower terminus has to dodge storms, the upper terminus needs to handle orbital correction, and everything in between needs to be moved out of the way of space debris and satellites -- and that's just to keep the cable intact. If you want to stick a permanent crawler in the middle of that and absorb the additional complexity and expense of transiting things from one end of it to the other, that station is going to have to move with all of that, which does not bode well for keeping constant tension on the lift cables.
Most plans (such as they are, still so speculative and requiring Handwavium production to get further) assume a bedrock-anchored base and engineering a way to resist any likely equatorial weather events. Those that don't may employ an
oil-rig-like bottom-end (on land, I've seen Inverted World-like railtrack-crawlers, overlaid upon the suitably prepared terrain) to give 'wiggle', but it's a near glacial movement compared with the 'ideal' movement of hundreds of miles, to skirt to one side of a storm's predicted path, within a handful of days, of something likely much more massive than the loaded
STS Crawler. For the bottom end, 'dodge' is an optimistic description of what could be done.
At the middle and top end, though, you can thrust the inverted-pendulum sideways (or 'bow' it by pushing on the middle and letting it flex) to avoid (known!) fly-by threats that might be more proximate than there is comfort for. Relatively quick reactions (apply some lateral thrust, perhaps even by applying electrodynamic potential against the Earth's magnetic field) compared with the low-end, but don't forget how fast (unpredicted) space-debris can hove into view.
Still, with all that, the tension is unlikely to change quite that much. Off-centering the ground 'anchor' and oppositely off-centering the far counter-weight terminus (the whole tethered structure crossing over the 'ideal' vertical) is likely to be the greatest over-stress, and on a 36Mm (to GEO) plus an additional fraction of that (to counterweight) the total angle of deflection from 'normal' is unlikely to be that great, with (non-linear, but nearly so) proportionate increase in tension
max across the structure surely being not beyond the in-built safety factor (which should probably already be calculated upon this most extreme of 'normal' operations the dodging north below to avoid a storm, whilst dodging 'south' above to avoid a NEO).
We also should probably do a lot of cleaning up of the current/near-future cloud of orbital debris, too, in laying the 'groundwork' for our tether.
Everything up there is equator-crossing and, if not in the current orbit then almost all inevitably in a future one if it's stable enough,
will cross any reasonable safety-zone presently imaginable.
(Beyond the requisite 20-Minutes-Into-The-Future already required for construction of the Beanstalk, it's
anybody's guess what safety-factors, precautions and defenses will be both necessary and possible to implement.)