If a creature can squeeze through a space, it can stop there too
Page 192 of the PHB says:
A creature can squeeze through a space that is large enough for a creature one size smaller than it...While squeezing through a space, a creature must spend 1 extra foot for every foot it moves there, and it has disadvantage on attack rolls and Dexterity saving throws.
Attacks in 5e are distinct compared to movement, that is a creature can move, stop to attack, then move again, but never attack while moving. A creature that is squeezing can still make attack rolls, although they are at disadvantage, so we can infer that the creature must be able to stop in order to make that attack. Since we now know that the creature can stop for an attack, and since there is no other specific rule saying that the creature cannot stop while it is squeezing, then we can infer that in general a creature can stop in any space that it can squeeze into.
Additionally, there is no rule saying that a creature has to end their turn no longer squeezing, nor that they can't end a move while squeezing. This is fairly easy to see in practice if players ever have to squeeze through a narrow cave passage that is longer than their move speed. If a creature couldn't finish their move while squeezing it would be impossible to, say, squeeze through a 200' long narrow cave passage, even though there is no rule that says this is impossible.
A Large creature can fit into a space as small as 5' x 5', though it will be squeezing while it remains there:
As stated before:
A creature can squeeze through a space that is large enough for a creature one size smaller than it
Page 192 of the PHB lists the various size categories. Large creatures are 10' x 10' large. Thus it can squeeze, and then stop moving, in a space large enough for a medium creature, which is listed as 5' x 5'. Though, if the creature does not have at least 10' x 10' of room available, it suffers all penalties associated with squeezing. So, a large creature can finish movement in all 4 cases that you have provided, though it will suffer all penalties associated with squeezing if it does not have at least 10' x 10' available.
Amorphous creatures don't typically have to squeeze
As Icyfire's answer illustrates, the rule for amorphous creatures follows:
Amorphous. The ooze can move through a space as narrow as 1 inch wide without squeezing.
The ooze can move into a space as narrow as 1" wide and finish moving there without squeezing. Since it isn't squeezing, it suffers none of the associate penalties (disadvantage on attack rolls/dex saves and more expensive movement).
Moving one square vertically will trigger an opportunity attack.
The relevant general rule is simply:
You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
(PHB, p. 195, "Opportunity Attacks").
And the rule for figuring ranges on a grid, which includes reach, is:
To determine the range on a grid between two things ... start counting squares from a square adjacent to one of them and stop counting in the square of the other one.
(PHB, p. 192, "Variant: Playing on a Grid")
D&D is an exceptions-based system, meaning that general rules apply unless there are more specific rules in place, and there's nothing in the single paragraph of rules for flying in combat (PHB p. 191) that change how these rules would operate.
So, if C1 can make melee attacks against C2, they are within one square of each other -- that is, they are adjacent on the grid. If C1 moves so they are more than one square away from C2, C2 can make an opportunity attack (and vice-versa). On a grid, all movement is in increments of one square, so any move from either combatant that puts more than one square of distance (making them no longer adjacent) between them is sufficient.
How big is a vertical square? The rules just say that a square "represents 5 feet". There are no specific rules for how to use a grid for three-dimensional combat (and remember that the rules for a 2D grid are themselves both optional and lightweight). The most direct interpretation of the rules is that a square on the grid represents a 5' cube of space. Thus while an active, flying, Medium creature will, strictly speaking, have bits of them sticking outside of the boundaries of "their" cube, what really matters is that any creatures who occupy any of the 26 adjacent cubes are within their one-square (or one-cube) reach.
Best Answer
You set the trap and it will trigger if the plate gets stepped on
Nothing says the space where you set the trap needs to be unoccupied.
However, it also doesn't say that entering the 5 foot area the trap is within or being within the 5 foot area activates the trap necessarily.
(emphasis mine). As such, number 3 is the closest to the correct answer if and only if the creature steps on the pressure plate when "it moves on the square later on." If the creature is smart enough to avoid the pressure plate in the center then it won't activate.