I think the answer to this would lie in the definition of "ability." I don't have my books in front of me, so I can't say for sure, and walking up 2 flights of stairs to get them doesn't strike me as a great plan, sorry.

All in all, I would think that it wouldn't.

Context?

I'm not aware of any item so specifically limited. If it does exist, it strikes me as more an example of bad design than anything else.

Oddly enough, the circlet of persuasion only gives it's bonus to charisma related checks. The difference between it and say a cloak of charisma is very important to Sorcerors and Bards.

No. According to the glossary an ability check is definied as a check of 1d20 + appropriate ability modifier. Grapple check is a type of opposed check, using BAB + 1d20 + Str mod + siz bonus + any other bonuses (say from feats). Attacks are definitely not ability checks.

I agree. Grapple and Attack rolls are not ability checks.

Items that add only to Strength-based skill and ability checks, however, strike me as ... unusual. I just have difficulty seeing something that makes you situationally stronger.

Aside from a lever. "This is my lever of might +2. Fear it."

I think the way to figure that out is to compare the price of the item to, eg, gauntlets that give a +4 bonus to strength+and item that gives +4 to dex (which would give the effects you're talking about) if the prices are the same then it's the same thing, if not then it's not.

Using the closest example I can find by a quick perrusal of the DMG, a circlet of persuasion(+3 to charisma checks) costs 4500. Cloaks of charisma are 4000 for the +2 and 16000 for the +4(36000 for +6)[Gloves of dexterity follow the same progression so it seems adequate as a benchmark]. So as an estimate(totally made up by a vague look at the progression-and probably wrong, my maths is not the best) a +3 charisma item would cost maybe around 9000. This would appear to mean that you don't get the same level of advantage from an item that modifies checks, and in the case of strength if the idea that it worked on grapple etc you'd only lose out on carrying capacity....not worth a difference of 4500 GP.

That's a great method for determining this. I agree.

The PHB (p.306) defines a check as something "other than an attack or saving throw".

An item that grants a +2 to strength (or any other ability, for that matter) would NOT grant a +2 to attack rolls. It WOULD though, give you a +1 to attack rolls, since you would gain a +1 bonus for every 2 points of an ability score.
(A strength score of 18 grants a bonus of +4. With a magical item that gives a +2 to strength would boost the score to 20, and therefore a bonus of +5 to all attack rolls, as well as any skill check that uses strength.)

Ok, after reading your post again, I realize that I don't think I read it right before commenting.
An item that grants any bonus (in your example, strength or dexterity +2) I would think affects ability checks. If you wear a ring that increases your dexterity, then not only would your AC increase, but also ranged attack rolls, skill checks using DEX (such as 'balance'), as well as any dexterity ability checks.
I hope this makes more sence, and is more helpful. I'm tired.