For example, can a wizard ready an invisibility spell with a range of "touch" and trigger it later on while not touching the target?
No, you can only release the spell if/when the target is within reach
When you ready a spell, you don't actually release it at that point in time, you cast it and release it later when your ready trigger is activated, so you need the target to be in range if and when that trigger condition is met.
When you ready a spell, you cast it as normal but hold its energy, which you release with your reaction when the trigger occurs. (PHB, pg. 193)
Furthermore, in the Range section of Casting a Spell, it says:
Some spells can target only a creature (including you) that you touch. (PHB, pg. 202)
My reading of these two rules combined is that you only target a creature when you release the spell, not when you cast it with a ready action. Ordinarily, when casting a spell normally, these two things happens as one, which is why it doesn't usually matter.
There's also this other quote from the Range section, but I do not believe this contradicts what I've said above:
Once a spell is cast, its effects aren't limited by its range, unless the spell's description says otherwise. (PHB, pg. 203)
Where it says "Once the spell is cast", I read this to include being released as well, so if you cast the spell as a ready action, technically you haven't finished casting it until you release it, so this rule doesn't come into effect until after the readied spell has been released. In other words, the target doesn't have to be within arm's reach to remain invisible once you've released the spell.
No, because the target isn't in range.
After readying a spell, you release it when the trigger occurs. However, you must meet all requirements of casting the spell normally (a clear path to the target, range, etc).
PHB, Chapter 10 - Spellcasting, Casting a Spell - Range, pg. 202:
The target of a spell must be within the spell’s range.
You might be confused by the sentence in the next page.
Once a spell is cast, its effects aren’t limited by its range, unless the spell’s description says otherwise.
It means in your case that after you cast Invisibility the target can get as far away from you as he can and the spell will keep its effects, even across dimensions because Invisibility doesn't have a restriction in range after casting like Faithful Hound that ends if move away more than a 100 ft. from it.
When you ready a spell, you cast it as normal but hold its energy, which you release with your reaction when the trigger occurs. (PHB 193)
Being in range is a requirement of casting a spell, which you clearly do with your Ready Action here, as are other necessities, like performing components. When your Action is done, the spell is cast. This is also signified by the caster spending the spell slot (if needed).
After the spell is cast, you do not have to be in range just as much as you do not have to perform components again.
This is all nice, but can lead to some situations that are hard to imagine. Your example of invisibility is fine, but what of, say, fire bolt? If you ready that against a target and it moves out of its range, can you still release it?
The problem is that the books do not define properly what "hold its energy, which you release" (PHB 193) actually means. Interpretation in the given circumstances will fall on the DM.