Yes, Bigby's hand can move in any direction even while grappling
Jeremy Crawford has clarified:
Q: is Bigby's Hand actually flying e.g. is it's movement speed = flying speed? The spell doesn't seem to describe it as neither "walking" nor "flying"
A: Bigby's hand doesn't have a speed. As the spellcaster, you can
magically move the hand in any direction.
So, all other issues aside, you can indeed move the hand into the air.
Bigby's Hand has a specific ability, Grasping Hand, that allows it to grapple creatures.
The hand attempts to grapple a Huge or smaller creature within 5 feet of it.
Barring any specification from the spell, there is no reason to think that this grapple does not follow the rules of grappling which say:
When you move you can drag or carry the grappled creature with you your speed is halved
Thus, there is no reason why the hand could not drag a grappled creature.
The hand is not affected by the grappling speed penalty
Bigby's hand is not a creature, it is a spell effect which has the ability to grapple. As clarified above, it does not have a speed. Since it does not have a speed there is nothing to penalize and that part of the grappling rule cannot affect it. So, the distance you can move the hand (60 ft) is not at all affected by the grappling speed penalty.
Consider this: would you reduce the distance a creature could teleport if it had reduced speed? No. Because that movement is magical movement specified by the spell effect and unrelated to speed in any way. The same exact concept applies to the hand.
Yes the creature can attack the caster.
Bigby's Hand: You create a Large hand of shimmering, translucent force in an unoccupied space that you can see within range... It has a Strength of 26 (+8) and a Dexterity of 10 (+0). The hand doesn't fill its space. When you cast the spell and as a bonus action on your subsequent turns, you can move the hand up to 60 feet and then cause one of the following effects with it.
Interposing Hand: The hand interposes itself between you and a
creature you choose until you give the hand a different command. The
hand moves to stay between you and the target, providing you with half
cover against the target. The target can't move through the hand's
space if its Strength score is less than or equal to the hand's
Strength score. If its Strength score is higher than the hand's
Strength score, the target can move toward you through the hand's
space, but that space is difficult terrain for the target.
The creature can attack the caster with a melee attack if the creature has more than 26 Strength therefore can move through the hand's space (difficult terrain for the creature) to get in 5ft (10 with a reach weapon) of the caster. The hand will still provide the half cover giving +2 bonus to caster's AC. If the attacker can attack from farther away however the hand can only grant the cover bonus. The reach has to be greater than 10ft to avoid the hand as it's a Large hand and covers 10x10ft square
If the caster would attack with a melee weapon however, the creature can't take advantage of the half cover as the hand specifies one way advantage (bold for emphasis on description).
Also any creature not targeted by the interposing hand will have no trouble getting close or attacking the caster.
Best Answer
That's what Bigby's Hand does when used in Interposing mode.
There is nothing preventing the incoming character from attempting to get around the hand - teleportation effects will do the job. Sure the hand will automatically move to interpose and still grant cover, but teleportation is not moving through the hand's space and not subject to the Strength issue.
Note that the hand itself is a large object, filling (albeit not completely) a 10×10 foot space. It prevents the target creature from getting any closer than that, but it doesn't stop it from attacking the Hand's caster with a ranged weapon or sufficient reach - it just gives half cover.
Probably the most important bit is that Interposing Hand only works against one target. It doesn't fill the space, so any creature other than the designated target can walk right through it. It's also a Concentration spell, which means hitting the caster hard enough or frequently enough can cause the spell to drop entirely.
Why would one use Grasping Hand instead?
Grasping Hand can grapple the target. A grappled target cannot move. Huge creatures often have reach, so the size of the hand may be insufficient to keep all creatures at bay - plus, those creatures are likely to be the ones with sufficient strength to power through Interposing Hand. Lock a creature down with Grasping Hand, and walk away.
And hey, you can crush them, too.
Why would one use Clenched Fist instead?
Clenched Fist and Grasping Hand both do damage, right? Sure... but Grasping Hand deals bludgeoning damage (a frequently resisted damage type) and Clenched Fist deals force damage (a rarely resisted damage type).
Why would one use Forceful Hand instead?
For the laughs, of course. Doesn't the idea of shoving people off of things with a ten foot hand make you chuckle?
TLDR
Bigby's Hand is a highly flexible spell. You have to pick the right mode for the right opponent.