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.
This actually goes all the way back to the first OD&D supplement, Greyhawk. The maximum spell level a Magic-User could cast was now limited by his Intelligence. Although interestingly, Clerics were explicitly not limited by Wisdom. The justification was that, unlike Magic-Users, a Cleric's spells were divine gifts, not based upon their skill. Intelligence also limited how many spells the Magic-User had.
Speculating, this was probably partly simulationist and partly for mechanical reasons. Arcane magical ability is tied intelligence, both lore-wise and in the the Magic-User's prime requisite was Intelligence. High level spells are more complex (the spell level system is a direct mapping from Chainmail's complexity system) so there's logic to a smart Magic-User being able to handle more complex spells than an average one.
Your Ability Scores had little mechanical impact in OD&D, primarily a bonus to experience gain. Greyhawk began increasing the existing, minor bonuses and penalties and adding new ones. Intelligence affecting Magic-Users spells was part of this increasing affects from Ability Scores. And only Magic-Users with an Intelligence below 11 were actually losing anything compared to the core game, because the 7-9 level spells were added with Greyhawk, and did not exist before ability limits.
This Ability Score limit spread to Clerics with AD&D. High level spells required a certain Wisdom and a low Wisdom could cause spell failure.
You may have noticed that Ability Scores were only limiting the casting of high level spells. As KRyan touched on, this is because of Prime Requisites / Ability Score requirements. RAW, you couldn't even play a caster with a below average Ability Score in their Prime Requisite.
So there's the historical precedent, which explains where it comes from. Was there any other reason to carry it forward, beyond tradition? It makes your Ability Score have a greater effect on your casting. The fact that spell strength is often not based on your Ability Score seems like it could be unbalancing to ignore the Ability Score requirement. I doubt it would be that bad, but I'd ask a optimization expert about it.
I personally don't have a problem with the idea that magic is too complicated for the average (Ability Score 10) person to grasp, and the smarter/wiser you are the more complicated spells you can comprehend and harness. It seems intuitive me, but if it doesn't to you and your players, I say house-rule away.
It is not allowed by the rules.
Your DM could allow it, of course, but that's a separate matter.
You can't buy an ability score lower than 7 for the same reason you can't buy an ability score higher than 18: it's not an option. The table only has costs for values from 7 to 18. We can't even know for certain what ability scores outside that range would cost, since the changes in cost don't follow an easy-to-derive pattern.
Furthermore, it's something that shouldn't be allowed. It encourages players to over-optimize their characters, making too-dumb-to-breathe musclebound brutes and too-ugly-to-look-at spellcasting geniuses. You end up with characters that are stronger than the game is balanced for in a few areas and utterly useless in others. Any DM who allows characters to purchase scores outside the 7-18 range deserves the dysfunctional party they'll end up with.