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).
You always use the DC of the spell, unless said otherwise.
You don't have to recalculate your con bonus, calculate your HD or anything, simply use the spell save DC for any abilities that require a saving throw.
From Transmutation (Polymorph):
Each polymorph spell allows you to assume the form of a creature of a specific type, granting you a number of bonuses to your ability scores and a bonus to your natural armor. In addition, each polymorph spell can grant you a number of other benefits, including movement types, resistances, and senses. If the form you choose grants these benefits, or a greater ability of the same type, you gain the listed benefit. If the form grants a lesser ability of the same type, you gain the lesser ability instead. Your base speed changes to match that of the form you assume. If the form grants a swim or burrow speed, you maintain the ability to breathe if you are swimming or burrowing. The DC for any of these abilities equals your DC for the polymorph spell used to change you into that form.
Reading Elemental Body, we have the following:
When you cast this spell, you can assume the form of a Small air elemental, Small earth elemental, Small fire elemental, or Small water elemental. The abilities you gain depend upon the type of elemental into which you change. Elemental abilities based on size, such as burn, vortex, and whirlwind, use the size of the elemental you transform into to determine their effect.
Air elemental: If the form you take is that of a Small air elemental, you gain a +2 size bonus to your Dexterity and a +2 natural armor bonus. You also gain fly 60 feet (perfect), darkvision 60 feet, and the ability to create a whirlwind.
As we can see, there is nothing replacing the more general text from the Polymorph subschool. Meaning that we use the DC of the spell used to assume the elemental form.
This can be seen on other spells, like Form of the Dragon's Frightful Presence ("DC equal to the DC for this spell").
Best Answer
The druid retains any abilities that their animal form can use.
Wild Shape states:
Emphasis mine.
There's nothing that would prevent the druid from using Haste or Bardic Inspiration that has been given to them by someone else while in animal form, so they are able to maintain those effects and make use of them.