Yes. This is correct.
To answer you bullet for bullet
Yes. This is what these do.
Yes, this is a correct reading, if you go from 18 to 20, you can still increase to 22.
Yes*, most things in 5e stack, though sometimes things don't stack with themselves (this isn't a bonus though, it's an increase, it's different). However, this is probably rare enough that this should not be an issue in a campaign (an if a DM gives out more than one of these per PC it's their own fault if their game breaks because of it). *I don't have this item's text in front of me so this is barring a caveat in the description, which if it existed would likely preclude this particular bullet anyways.
So basically, yes. This is an item (among several and a L20 class feature or two) that breaks the normal rule for maximum 20 ability scores. It breaks it, and it does so permanently for the character.
RAW
From strict reading of the ASI, you shouldn't be able to, since it does not say "You can't increase an ability score above your current cap" or something on these lines, it just says "You can't increase [...] above 20".
Rules as (probably) Intended
Now, is it intended? Probably not.
In the PHB, the only way to go over your cap is through Barbarian's Primal Champion feature, which is at 20th level and you are not getting any more ASIs after that.
So one possible reading on this is that it's worded as it is for simplicity, since they were not considering the items on DMG.
The problem is that, from my experience, the DMG is written (more) vaguely (than the PHB), which might even be intentional since it's supposed to be read by the DM and he can decide whatever he wants. This means it's probably implied that increasing the maximum score by 2 (to 22) allows you to increase your stat score to 22 by any mean, including ASI.
A further indication that you should be able to do it is that every mean of increasing your stats says something on the lines of "up to 20" (AFAIK), instead of "up to your cap". This means any feature increasing your cap doesn't do anything, since everything is hard capped in its own text anyway (Ioun Stone as another example besides ASI).
For clarity, what I mean here is that there would be no reason to word it as "your maximum for that score increases by 2" instead of "up to any value" or "not hard capped at any value" or anything like this if it was not supposed to influence any other item or feature.
About your last question
In fact, is 20 even the maximum for your ability score when using items; the quote seems to only apply to Class Features.
The quote is a general rule. This shouldn't matter though, because as far as I know, the items explicitly state their cap (Deck of Many Things' star card caps at 24, Ioun Stones cap at 20, etc).
Best Answer
Yes, the Star card can permanently increase your ability scores beyond the normal limit of 20.
The Player's Handbook, on page 15 in the section about applying ability score improvements, states the following rule:
Note that, specifically, this does not say "you cannot have an ability score above 20"; it states that you cannot use ability score improvements to increase an ability score above 20 - the rule only has importance at the moment you are altering your stats.
Following standard "specific beats general" interpretation, the Star card offers you an ability score improvement which specifically clarifies that it can, in fact, raise your ability score above 20, up to 24. Having applied the benefit of the card, if your ability score is now 20 or higher, you can't improve it further with a normal ability score increase, as that's subject to the general rule that you can't increase the score above 20. But there's no rule that states that, if you have legally increased your score past 20 by some means, it has to go back down again. It's not really so much a "cap" as a threshold - it's difficult to cross but you're not forced back if you do it.