A creature making use of climbing movement, on a suitable surface, when interrupting its movement during climbing, can it act freely by remaining in the place where it is? Is it necessary to spend an action and/or occupy one or both hands while remaining motionless? Is there any risk of falling while immobile, if targeted by something/effect, even if the creature has climbing movement?
Requirements You have both hands free.
You move up, down, or across an incline.
The Climb action is only attempted when moving. No action is required to stay stationary on a surface, and the requirements here taking up both hands would only apply while taking the action (though letting go completely could cause a fall depending on the circumstance).
If you would be knocked prone while you're Climbing or Flying, you fall
Most effects don't cause a creature clinging to a surface to fall unless it moves them out of reach of the surface or knocks them prone. Using a reaction to Grab an Edge at the same DC as the Climb is a good way for those creatures to stop falling in those cases.