The answer is in your bold letters.

The subject can climb and travel on vertical surfaces or even traverse ceilings as well as a spider does. The affected creature must have its hands free to climb in this manner. The subject gains a climb speed of 20 feet; furthermore, it need not make Climb checks to traverse a vertical or horizontal surface (even upside down).

The description says that it doesn't need to make Climb checks in order to traverse a vertical or horizontal surface. In that case, the check is to overcome the slippery effect of Grease spell. I would assume that it is the same as saying that a creature with walking speed needn't make any Balance checks to walk on an horizontal surface. But it still needs to make them in order to overcome obstacles. Grease is an obstacle, not a surface.

The spell calls for a Balance check, but given the specificity of the situation you describe, your GM could perfectly call for a Climb check instead. It seems reasonable.

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.