[RPG] Does the half orc’s Savage Attacks trait work when wild shaped

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).

Yes, Divine Smite will work while Wild Shaped. Wild Shape allows you to use your class features:

You retain the benefit of any features from your class, race, or other source and can use them if your new form is physically capable of doing so.

And Divine Smite works like this:

Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one paladin spell slot to deal radiant damage to the target, in addition to the weapon's damage.

Divine Smite requires a melee weapon attack. Every beast type creature that I can think of has at least one attack that is a melee weapon attack. For example, the Wolf:

Bite. Melee Weapon Attack: +4 to hit, reach 5 ft., one target. Hit: 7 (2d4 + 2) piercing damage. If the target is a creature, it must succeed on a DC 11 Strength saving throw or be knocked prone.

So you can use your Divine Smite with the melee weapon attacks provided by your Wild Shape form.