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).
Rule interpretation in general
There are two points to remember when looking at an ambiguous rule:
Your group, with the DM as final arbiter, decides the ruling for the rule. So, you need to establish what this is for each group that you play with.
Specific beats general. Start with the most general rule you can find and see how the specific rules change this.
Armor resizing in particular
For your issue, start with PHB p. 145 (ignoring the optional part because it makes no difference to the case at hand)
Variant: Equipment Sizes
In most campaigns, you can use or wear any equipment that you find on your adventures, within the bounds of common sense. For example, a burly half-orc won't fit in a halfling’s leather armor, and a gnome would be swallowed up in a
cloud giant’s elegant robe.
There's your general rule: no armor that fits a PC race (size S or M) would fit a bear (size L).
Wild Shape (p. 67 PHB) says:
Worn equipment functions as normal, but the DM decides whether it is practical for the new form to wear a piece of equipment, based on the creature’s shape and size. Your equipment doesn’t change size or shape to match the new form.
No help there; the armor will be too small and the wrong shape. In addition, there is a ruling that has to be made right here: is it practical for a bear to use armor even if it is the right size and shape? My ruling would be yes but your results may differ.
Now from DMG, p. 140:
In most cases, a magic item that's meant to be worn can fit a creature regardless of its size or build. Many magic garments are made to be easily adjustable, or they magically adjust themselves to the wearer.
The ruling needed here is does the hide armor a) need easy adjustment or b) magically adjust itself. If b) then all is easy, if a) then the druid may need to allow it to drop to the floor and then get some assistance to put it on which would take 5 minutes (PHB, p. 146) which makes it pretty useless in combat.
Those are the issues; your group needs to make the ruling.
Best Answer
Yes, the rules are scattered. Nevertheless, let's see what we can find out.
A Druid does not gain the bonuses to Strength, Dexterity, Constitution and Natural Armor listed on the Changes to Statistics by Size table. This is because the table is presented in the specific context of improving monsters by HD advancement. It does not apply in other contexts, such as a Druid taking the shape of another creature, or even an Animal Companion gaining bonus HD. It could be said that the Druid gains the "AC/Attack" bonus or penalty listed on the table, but that is because that column is just calculated from a table that is applicable - the Creature Size and Scale table.
The latter table is not restricted in scope as the Changes to Statistics by Size table. If I may dare say it, it is only logical that it isn't - these statistics are as basic as "Space".
Deriving from that table, we get answers to your specific questions:
Note, however, that points 2-4 should already be factored into the statistics of the creature into which the Druid is Wild Shaping.