Spells do (only) what they say

So, from the point of view of the spell, answering your questions:

1. The spell doesn't say anything about them being easier or harder to hit, thus they aren't (necessarily).
2. The spell doesn't say anything about later checks or saving throws, thus they aren't forced.
3. Unclear.

So, purely from the spell, we can't say much. Everything else is...

Up to the DM

Now, how do we adjucate the fact that the character is hanging in something?

Our brain is used to usual gravity, so let's just think on usual gravity. As for when the character is hanging into the object, we could just think of a character hanging into a chandelier (most usual thing to grab in the ceiling I can think of). It also helps because this is a situation more likely to be experienced by players and DMs. I've had rogues actually jumping from objects (specifically chandeliers) in the ceiling to dodge enemies that were in the ground, for example. So, back to your question:

1. That depends on the object the character is hanging into and how exactly he is. We could approximate his condition as simply Grappled or, in a worse scenario, as Restrained.¹ Certainly changing the AC is not consistent with other rules - even paralyzed or unconscious characters don't lose AC, even if they are in light armor (that is highly Dex dependant).

   • For the object they are hanging, for the chandelier example I'd ruled that the rogue could dodge as usual. He had some freedom on where to grab the chandelier and could move using his hands. Personally I didn't think that restriction was enough to give him "disadvantage at dodging" (a.k.a. being attacked with advantage). If he was hanging in a pole-like object, though, maneuvering would be harder (I think, at least) and the disadvantage at dex saves and advantage for attacks made against him would apply.

2. That depends on how you rule climbing. Simply hanging is obviously easier than climbing. If you take into account their weight and carried weight in defining your climbing DC, do the same here. I think the most important thing is to be consistent. For me, I actually ignore every encumbrance rule from PHB :) - and from experience with other tables, it's one of the most ignored rules in the game. From that perspective, I would say the character can easily keep hanging in the object, if that's all he wants to do.

3. For movement, I'd say "climbing the ceiling" (like Spider man does) is not a thing, unless they have something that can help them with that. We can climb walls because we can actually grab the rocks in a way that it nullifies the gravity vector. Doing that in the ceiling would need the ceiling to specifically allow it somehow (holes or something like that). They could, however, move jumping from one object to another (think Prince of Persia?) and I would rule that it requires either Acrobatics (preferably) or Athletics (possible). Note that from my first point, doing so actually requires an Action, not Movement speed, as the MS is zero², meaning they can only jump once per turn.

I can't think of anything more illustrative for the ruling than the chandelier example. The fact that the gravity is reversed and now the floor is the ceiling essentially changes nothing, except for the objects that will be used.

Finally, I would note that objects in the ground, contraty to objects in the ceiling, usually are not anchored to the ground because they don't need to. That's the main reason I think it's easier to think in the usual situation rather than the reversed. In particular, I would like to emphasize the condition for being able to make the Dex ST:

A creature can make a Dexterity saving throw to grab onto a fixed object it can reach, thus avoiding the fall.

This objects needs to exist, i.e. an object that is fixed (anchored to the floor) somehow.

¹ About the Grappled/Restrained condition, I use them as approximations because they are the closest I can find inside the rules that fit decently with the described situation, either as conditions or as spell effects overall. You clearly can't "move" while grabbing something to not fall (grappled) (unless you get some foot support) and if the object you are grabbing onto is not a good support, you will have difficulty doing anything with either your body or arms (similar to how restrained works).

² While we have rules on Jumping from the Special Movement section, you aren't literally jumping, since you aren't even using your legs. Jumping with your arms is alot harder than with your legs. This is my reasoning to spend an action jumping from one object to another.

It seems to be that the Orb refrains from freezing the liquid if it is cast underwater.

The intention seems to be that throwing the orb underwater wouldn't be considered striking the water. Striking usually refers to as one thing coming into contact with another, with the intent to hit, and when your orb 'strikes the body of water' isn't clear. It is also IN a body of water, so the orb isn't really capable of striking a body of water if it is already in one.

If the globe strikes a body of water or a liquid that is principally water (not including water-based creatures), it freezes the liquid to a depth of 6 inches over an area 30 feet square

Emphasis mine.

While getting into a bit of semantics depth is the distance from the top of something to the bottom. It would be slightly unintuitive if one could consider something 'the top of' if it started in the middle of something, such as the ocean or body of water you are in. Depth doesn't seem to apply properly in this scenario.

If it froze underwater using area and depth would be an inconvenient way to do things.

If the intention of the spell was to properly freeze while cast underwater it would, for more ease and out of a normal expectation, probably freeze in a radius or a more spherical shape. The wording of the spell seems to indicate it is made to freeze the surface of water. It is measured in area and with depth included, not a measurement and description you would use for underwater (at least not normally).

With this and the phrasing of 'strike' and with the observation that the area the spell freezes is an 'area' (with a six inch depth) it can be argued that the orb would not instantly explode with contact with water, if it were already underwater. But further on...

It would not freeze any water.

The freezing of the water is not directly related to the cold damage the spell does. With the orb not properly preforming it's second function, the freezing of water, there is no reason that the freezing would take place. Cold damage does not freeze water without DM intervention (or a certain effect says so, like with the case of some spells like Wall of Water).

Lastly...

There is no difference between casting the spell or taking the orb and throwing it.

It shatters on impact, with the same effect as the normal casting of the spell. You can also set the globe down without shattering it. After 1 minute, if the globe hasn’t already shattered, it explodes.

Emphasis mine.

Outside of a slight word difference between 'explodes' and 'shatters' the spell functions the same whether you are throwing it or casting it and throwing it instantly as apart of the spell. There is no difference between the orb shattering or exploding in a body of water. As long as the orb doesn't strike the water it shouldn't be.

Perhaps more is needed....

If designer intent was that the spell was to instantly freeze water it hit (or started in), excluding water based creatures such as elementals, then it could be that this interpretation is incorrect. I've tweeted Crawford over the issue to see what the intent is.