The half-giant monster entry lists Powerful Build as a Special Quality. Alternate Form explicitly lets you retain those, so yes, a druid with Powerful Build does retain that bonus, which is sizable. Powerful Build explicitly stacks with things that change your size, so if the druid used Wild Shape to turn into a Large creature, he would count as Huge for the purposes listed in Powerful Build.
See here for a list of options for obtaining Powerful Build. Note that all of them have positive Level Adjustment, which the homebrew lesser half-giant you found does not. Many DMs would balk at that. In reality, Powerful Build on its own probably doesn’t justify LA +1, but Wizards evaluated it as such. Level Adjustment is bad in general, but truly awful for spellcasters. And the half-giant race has nothing else of use or value to a druid, though use of the Primordial Giant template from Secrets of Xen’drik, at least, would make the half-giant more caster-ly (though still nothing particularly relevant to druids).
Answering out of order, because #1 depends on #2.
- Does the Minotaur's Powerful Charge ability count as having the Powerful Charge feat; allowing him to take the Greater Powerful Charge feat?
Strictly speaking, no. The minotaur’s Powerful Charge is marked as an extraordinary ability, and not mentioned as a feat. Feats are extraordinary (unless they’re not), but bonus feats are not marked (Ex). Therefore, a minotaur cannot take Greater Powerful Charge without taking Powerful Charge.
It not being the same as the feat is dumb, in my opinion; I generally recommend houseruling it. Though, do note that the minotaur is only seeing a +1d6 bonus when, as a Large creature, the feat would give +2d6. It’s unclear to me if Greater Powerful Charge should add +2d6 (as normal) or +3d6 (i.e. +2d6 above what Powerful Charge normally adds). That’s up to the DM, but it seems weird to me that a minotaur would have less benefit from these feats than other creatures.
- Does anything prevent the minotaur from taking the the Powerful Charge feat?
No, the minotaur (strictly speaking) doesn’t have the it, so it can take the feat.
- Would using his Powerful Charge ability stack with using the Powerful Charge feats?
If it is separate, per RAW, yes. And since Powerful Charge (Ex) and the Powerful Charge feat are different sources, the untyped damage bonuses from each stack.
Again, this is dumb. But, then again, it does mean that a minotaur can take Powerful Charge and Greater Powerful Charge, gaining +4d6 damage on top of its native +1d6 damage, for +5d6 damage – more than other creatures get. That becomes a pretty strong case for treating the minotaur’s natural Powerful Charge as separate.
Conclusion
The fact that Powerful Charge (Ex) and the Powerful Charge feat are different is dumb, but it actually works out, allowing minotaurs to take the feats and end up a bit better at powerfully charging than other creatures, which makes sense for a creature naturally inclined to charge.
Best Answer
Powerful Shape does not affect a Natural Attack's damage.
Size-based special attacks are attacks that depend on your size for their effect, either to see if the ability can be used or not or to calculate their effects. If you take a look at the three examples given, we have the information we need:
All three have very similar wordings on how they affect creatures smaller or of similar size category than the creature using the ability. There is a comparison of creature sizes on those abilities. This wording also appears on Whirlwind, Pull, Push and Rock Throwing.
If we look at the Natural Attack ability, nothing in its description says it behaves or is restricted in any way based on your size against the target's size. Only that your size affects the damage, regardless of the target's size. So Natural Attacks are clearly out of the scope of Powerful Shape.
If it did, it would be something really important to leave out of the feat's description, as there is clearly another feat that increases a Natural Attack's damage.