The levitate spell contains the following text:
...you can use your action to move the target, which must remain within the spell's range.
The spell's range is 60 feet.
Levitate doesn't seem to provide any other major stipulations, so in theory, if I am levitating a target that is 60 feet away from me in one direction, can I move it 120 feet over me to the 60-foot extreme in the opposite direction, as a single action?
No, as you are only able to raise or lower the target, and the limit to this is 20ft per round. Nothing is mentioned in the spell description of being able to move the target horizontally.
You can change the target's altitude by up to 20 feet in either direction on your turn. If you are the target, you can move up or down as part of your move. Otherwise, you can use your action to move the target, which must remain within the spell's range.
I've always understood it as:
You can use your action to move the target (up or down), but must remain within the spell's range.
What I think you're missing is this relevant constraint:
Levitate
... The target can move only by pushing or pulling against a fixed object or surface within reach (such as a wall or ceiling)
This quote suggests that Levitate and the subsequent actions as you concentrate on the spell only allows you to move the target vertically. The target can only move horizontally if they can "climb" horizontally.
This has to be the case, otherwise the spell would be the same as Fly.
No.
The spell only permits the caster to vertically move the target. This makes it a great lockdown spell against a melee-only opponent as they can be lifted into the air and subsequently pincushioned, provided there isn't anything for them to grab onto.
The only form of lateral movement permitted by the spell is detailed in the spell's second paragraph which states that the target must push off of a fixed object within reach and when doing so the creature would use its climb speed to do so.
If the target were to elect to 'climb' as described in the second paragraph, they could potentially do so provided they remain within the 120' diameter sphere around the caster you described in your question. However, the caster cannot force them to move laterally and would need to rely on alternative methods to push the target around, perhaps a warlock ally with Repelling Blast.
The spell's restriction on range limits how high you can lift them up away from you: 60 feet. Usually close enough to target you with ranged weapons if they have them. It also determines when they escape from the spell if they climb away, since once they get outside of the 60' radius, the spell will end for them.
Per the spell's description:
You can change the target’s altitude by up to 20 feet in either direction on your turn.
That's the movement you get. It's not lateral, but vertical.
