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).
To my mind, the overriding rules here are these:
Alternate Form
The creature retains the type and subtype of its original form.
Undead Type
You cannot have an undead that has a Constitution score, so since Alternate Form has you retain your type, you cannot gain a Constitution score. I believe this is “more specific” than the general rule that you gain the target creature’s physical ability scores when using Alternate Form.
You could argue the opposite way, however, that Alternate Form is a more specific case than the general rule that undead do not have a Constitution score. But note that this results in a living (has a Constitution score) undead (retains that type), which to me is a good enough reason to side the other way.
As for using Charisma in place of Constitution, this is a common feature for undead, but not a universal one. In some places, that privilege requires a feat (e.g. Undead Meldshaping in Magic of Incarnum). Depending on your build, I might want to tax you for it; Cha-to-everything is a fairly potent strategy as it is, and undead is a desirable type anyway. Getting to use Charisma for yet more things, and avoiding the “weakness” of the undead type’s lack of Constitution, might be problematic. But if you weren’t already pursuing a Cha-to-everything build, I’d probably give it freely.
Best Answer
Well, the answer to the previous question is up to the DM. Obviously, anything that follows up from "the DM allowed it" will also be up to the DM to rule.
So, splitting your question.
Do they need to make a check?
I would usually say that every action needs a check, unless
I doubt anyone will be ruling it as too hard. So, the question is: is it easy enough to not require a check? - We will be back to that when discussing the DC.
What check is it?
The most common check for this scenario would be Athletics (Strength) or a Strength Saving Throw. It is the check used for grabbing something and hanging on it against an opposing force.
An example of the saving throw being used is given in Lost Mine of Phandelver
What DC?
It heavily depends on the scenario. I have usually ruled that characters could hang themselves in objects while not moving without effort - so my following answer is consistent with that.
If the spider is moving slowly, it should be easy enough and not require a check at all. If it is moving normally or quickly, it should be more challenging to the character. The easiest translation I can think of is using the same DC you would use for climbing that wall anyway - the advantage of using the spider is that it does not slow his movement speed by half.
For characters being attacked, the closest I think we have in the rulebooks is the Concentration mechanic, which states
it would be more a Str saving throw, though, than a Con. Note that being attacked while mounted does not dismount you in any way - so, by the rules, you can as well simply ignore the attacks (for the mounting process).
For the ceiling scenario, it would probably be very hard or even impossible to stay on the spider's back without any kind of extra support.