Yes, creatures can attack anything within their natural reach

As you pointed out,

Creatures that take up more than 1 square typically have a natural reach of 10 feet or more, meaning that they can reach targets even if they aren't in adjacent squares.

Unlike when someone uses a reach weapon, a creature with greater than normal natural reach (more than 5 feet) still threatens squares adjacent to it.

So what does "threaten" mean?

You threaten all squares into which you can make a melee attack, even when it is not your turn. Generally, that means everything in all squares adjacent to your space (including diagonally).

As a visual illustration that natural reach doesn't exclude adjacent squares (or just to visualize reach in general, especially diagonal reach) these graphics can be helpful as well.

Rules Compendium p. 150 (emphasis mine):

A reach weapon is a melee weapon that allows its wielder to strike at targets that aren’t adjacent. Most reach weapons double the wielder’s natural reach, allowing the wielder to attack at that reach but not within its normal reach. A typical Small or Medium wielder of such a weapon can attack a creature 10 feet away, but not a creature in an adjacent square. A Large wielder wielding a reach weapon of the appropriate size can attack a creature 15 or 20 feet away, but not adjacent creatures or creatures up to 10 feet away. Tiny or smaller creatures gain no advantage from reach weapons.

Rules Compendium p. 151:

A wielder gains no reach from a reach weapon that is too small. No additional reach is granted by a reach weapon that is too big.

Based on those two things, we can answer your two questions:

1. No. Tiny creatures threaten no squares around them, and reach weapons don't benefit them. As a larger reach weapon doesn't convey any extra bonus, they don't get any reach no matter how big the weapon is. (Though the mental picture of a tiny creature using a gargantuan halberd is pretty funny.)
2. If the weapon is inappropriately small, they do not gain a reach boost at all. If it's inappropriately large, they do not gain an extra reach boost, but they would gain the normal reach boost for a creature of their size using a reach weapon. So a Dwarf using a large spiked chain gets the same reach as a Dwarf using a medium spiked chain.

I think your proposed feat is going to work as you intend. A fine spiked chain won't convey any reach, it'd just be like using a melee range weapon only doing a lot less damage.

Specific Q/A

Basic Question: Does a grig (MM 235) wielding a Tiny longspear (PH 116, 119) (2 gp 5 sp; 0.9 lbs.) threaten no squares, adjacent squares, or squares 10 ft. away?

The grig is tiny, using a reach weapon. It therefore threatens no squares.

Then: What about a grig wielding two-handed a Small spinning sword* (Secrets of Sarlona 137-8) (50 gp; 3 lbs.)?

Threatens no squares. The spinning sword definition says that it's a reach weapon, and tiny creatures do not gain reach from reach weapons.

Then: What about a human wielding as a light weapon a Tiny longspear?

Threatens his natural reach only. Undersized reach weapons convey no reach, but they don't take away your natural reach (since said human could just punch that square).

Then: What about a human wielding two-handed a Large spinning sword (100 gp; 6 lbs.)?

Yes, the human would threaten squares 10' away, and 5' away (as the spinning sword works at close range, like a spiked chain).

Does The Rules Compendium Contradict The SRD/DMG?

No, it doesn't. The SRD's reach definition is as follows (emphasis mine):

Glaives, guisarmes, lances, longspears, ranseurs, spiked chains, and whips are reach weapons. A reach weapon is a melee weapon that allows its wielder to strike at targets that aren’t adjacent to him or her. Most reach weapons double the wielder’s natural reach, meaning that a typical Small or Medium wielder of such a weapon can attack a creature 10 feet away, but not a creature in an adjacent square. A typical Large character wielding a reach weapon of the appropriate size can attack a creature 15 or 20 feet away, but not adjacent creatures or creatures up to 10 feet away.

Tiny creatures have a natural reach of 0. If you double 0, you get 0. Thus while the RC definition is lengthier, it doesn't contract what the SRD or DMG say.

The only thing in the SRD about oversize/undersize reach weapons is the bit in the above quote about using a reach weapon of the appropriate size. The RC expands on this, but again doesn't contradict what is said. (You could argue that in the SRD, an oversize reach weapon also doesn't grant reach as it's not "appropriate", but they didn't really spell out what they meant so I don't find the RC version problematic.)

Can A Tiny Creature Ever Get Reach?

Yes, as the DMG mentions:

Tiny, Diminutive, and Fine creatures have no natural reach. They must enter an opponent’s square (and thus be subject to an attack of opportunity) in order to attack that opponent in melee unless they are armed with weapons that give them at least 5 feet of reach

That makes it clear that tiny creatures are not forbidden from getting reach, if they have some way to get it. Some examples of ways to do that:

• Shadowstrike (MIC) - On a weapon, activated to give that weapon +5' to it's reach. As it's just a flat addition to reach, it works even if the creature has 0' reach normally.
• Lunging Strike (PHBII) - Feat that allows you to make an attack with +5' to it's reach. Again this is a straight addition, so it works even if you had 0' reach.
• Inhuman Reach (LoM) - Feat that increases your character's natural reach by +5'.

What About Whips?

Whips have a different wording than the other standard reach weapons, as shown here:

The whip is treated as a melee weapon with 15-foot reach, though you don’t threaten the area into which you can make an attack. In addition, unlike most other weapons with reach, you can use it against foes anywhere within your reach (including adjacent foes).

Compare to the Spiked Chain (another reach weapon that works in melee range):

A spiked chain has reach, so you can strike opponents 10 feet away with it. In addition, unlike most other weapons with reach, it can be used against an adjacent foe.

The "X has reach" wording on the chain is the same as you see for every reach weapon in the SRD, except the Whip. The first part of my answer covered how those work with larger and smaller creatures. But is a whip special?

1. The strictest possible reading says that a whip is treated as a melee weapon. It also happens to have a 15' reach, and that number is specific. Does it change for larger or smaller creatures? The rules don't say. If you wanted to do a very strict reading, it's always 15', no matter the creature size. This starts to make no sense with gargantuan or larger creatures as it actually lowers their natural reach, and would give tiny creatures 15' reach with a whip and 0' reach with any other weapon, so it doesn't work very well at the table.
2. A less-strict reading would instead say that a whip is a reach weapon (that can also strike in melee, like a spiked chain) that adds triple to the natural reach, instead of double for normal reach weapons. That means a whip scales up for larger creatures like other reach weapons do (except farther), and tiny creatures get no reach at all when using one. This has the virtue of being consistent with how reach weapons normally work. Also working in the favor of this one is that the reach weapon definition mentions whips as a reach weapon.

So for whips, it comes down to how much stock you put into the odd wording compared to other weapons. I can't give a reference for what the writers actually intended to do there, as one doesn't seem to exist. I do know that the second option is far more consistent in how it plays, so I would certainly favor it in actual play.