There is not an exact name for the fear of low places. However, there is a fear of jumping from high and from low places. This phobia is called Catapedaphobia.
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I'm not a psychologist, but I'm thinking it would be "hypoterraphobia" (hypo = below, terra = earth or ground)
No, there isn't a phobia of going underground, but the closest thing would be "Rupophobia", the fear of dirt.
The fear of going under things is a form of claustrophobia, because you are partially enclosed when something is above you.