Yes.
- The spell greater invisibility says its effect is just like the spell invisibility.
- The spell invisibility says it makes you invisible, the condition.
- The condition "invisible" says to see the ability "invisible".
Note that each step along a chain like this may introduce changes. For example, if any details of a spell contradict the details of the condition it's based on, the details of the spell override the details of the condition.
For a trivial example in this case, the invisibility spell adds the information that the condition ends when the subject attacks, while the ability implies that an invisible attacker can stay invisible: the spell overrides that. (And of course, greater invisibility overrides this part of invisibility again, making the subject stay invisible after attacking.) As another example, greater invisibility and invisibility can be detected by detect magic, while the invisibility ability may or may not be so detected, depending on what type of ability it is in the creature description.
1. Radius Patterns follow Grid Lines
The ring version of Wall of Fire says this about its area of effect:
a ring of fire with a radius of up to 5 ft. per two levels
If you turn to the back of the DMG (p. 307) or look at this answer, you can see some patterns for radius spread effects. All of them spread along square edges. They're a ring of squares, rather than a circle. In this pattern, a square can't be half in or half out, because it only travels along the lines.
(This is actually also true for line effects, as PHB p. 176 shows. If a line goes through any part of a square, the entire square is affected. For something like Wall of Fire, an entire square has to be in one state or the other to follow with this, but if you place it along the grid lines it's fairly easy to do that.)
2. Yes
As you noted, Blade Barrier doesn't give a thickness or anything to imply that it's more than a curtain of blades (in fact, it says its a curtain of blades). You can treat it the same way as Wall of Fire: draw it on the grid lines. (The rules for area of spells mention that you should draw it starting from a grid intersection, so you can follow the grid lines.)
Regardless of the shape of the area, you select the point where the
spell originates, but otherwise you don’t control which creatures or
objects the spell affects. The point of origin of a spell is always a
grid intersection. When determining whether a given creature is within
the area of a spell, count out the distance from the point of origin
in squares just as you do when moving a character or when determining
the range for a ranged attack. The only difference is that instead of
counting from the center of one square to the center of the next, you
count from intersection to intersection.
This also makes the cover part of it easy to handle: anything on one side of that grid line attacking the other side has to deal with the cover portion of the spell.
3. Creature Size Matters
If you're placing it on grid lines (as you should), then yes, you can't actually place it on top of a medium creature. Large (or bigger) creatures take up multiple squares and you could have the barrier cast such that it appears in the middle of one. That makes this part of the spell make sense:
If you evoke the barrier so that it appears where creatures are, each
creature takes damage as if passing through the wall. Each such
creature can avoid the wall (ending up on the side of its choice) and
thus take no damage by making a successful Reflex save.
Medium or smaller creatures wouldn't be damaged, as the barrier is not "where the creatures are".
4. Because the spell says so
This point has no rules-as-written answer, except because the spell says so. :)
I'd speculate that the game designers did that with the reasoning that as the spell comes into existence, someone can react and get out of the way before it can fully attack. Whereas once it's already there, someone has to go through it. But I don't know of any rules, anywhere, that explain why they did it this way.
5. Yes, you can attack
Blade Barrier says this:
Any creature passing through the wall takes 1d6 points of damage per
caster level (maximum 15d6), with a Reflex save for half damage.
By a strict reading, passing through the wall is movement. By that reading: yes, you can attack through it without taking damage. I'm not aware of any rules clarification on that point.
Anybody attacking through it does have to deal with cover:
A blade barrier provides cover (+4 bonus to AC, +2 bonus on Reflex
saves) against attacks made through it.
Best Answer
Attacking with an active Ring of the Darkhidden does not break the limited invisibility it grants.
Nothing in the ring's description says that attacking breaks the limited invisibility it grants. Importantly, it does not include the phrase "as the spell" (thanks @fectin for mentioning that) common to a lot of "spell in a can" magic items - notably including the Ring of Invisibility, which the wearer can activate to "benefit from invisibility, as the spell".
The "invisible" status assumes that attacking does not break invisibility:
The spell invisibility explicitly ends if the creature it renders invisible attacks, further cementing that "attacks while invisible end invisibility" is a specific "feature" of that spell, rather than of invisibility generally.
Since "attacking breaks invisibility" is a feature of the spell invisibility (and a few others that are "invisibility, but..."), there's no particular reason to believe that attacking while wearing the Ring would render the wearer visible to darkvision.
"But", you say, "the item is based on the spell invisibility! Surely that means that attacking ends invisibility." The requirements on magic items serve a couple of purposes (eg., gating who can make the item - a Ring of the Darkhidden requires invisibility, which is an arcane spell; clerics would have a hard time making such a ring). However, the effects of the spell and the effects of the item aren't firmly linked - see the Ring of Communication (also on p122 of the Magic Item Compendium), which requires detect thoughts, but grants that:
The Ring of Communication does not grant telepathy of any sort, instead relying on spoken words, even though the base spell is detect thoughts - one of the earlier telepathy spells.