Looking at his wiki... Is his damage really that low? 58 physical/magic damage on his Q, and only 58 physical on the first hit? Why? Is his hybrid damage that strong? Is it just to buffer his auto-attacks? Is Sheen mandatory on him?
His damage early game is more than it would appear, and he can duel decently. It will fall off a decent amount, but he'll always do at least some.
He has the highest base AD in the game, so yes, you need a sheen. You will eventually always want one in your build, although you don't always have to rush it.
How much damage is acceptable on him?
This depends on the game, and your role in it. Oftentimes,, you'll be pretty tanky, with only a sheen item. Other games, you can afford to get a triforce and one other damage item (such as a damaging jungle enchantment). You'll almost always need a few tank items though, because you're not going to be bursty no matter how much damage you build.
Why does his W need to ramp up before he goes anywhere quickly? I don't understand this change to him, it just SOUNDS like it feels bad to use.
It does feel bad. You'll deal with it.
What's his 'optimal' build most games?
You have a few options. The first option is the initiation build. This is the most common Skarner build, although I personally think it is outclassed by other champions. For this, you build a Juggernaut jungle item, then follow up with a Righteous Glory and Iceborn Gauntlet. Your boots must be mobis or swifties. From there, build tank. You will do minimal damage with this build, so your goal will be to get a good pull, then peel for your carries with your E. If you can't make a pull, then this build will not give you many options for helping your team.
Another build focuses slightly more on early damage, building tanky items later. With this build, you get either a Warrior or Devourer enchantment, get a sheen, then build either a triforce or Iceborn Gauntlet depending on the needs of the game (a highly melee and AD enemy team should call for the Gauntlet). Then, build very tanky. Boots are your choice. With your high base stats and your shield, you should be able to survive the early game even without rushing health. A Zephyr can eventually be useful in this build, although I wouldn't recommend it in most games.
With any build, keep in mind that you have very high base health and a shield, so you benefit more from resistances than from flat health (you still need a little, but not much). You also benefit a lot from CDR, so try to max that.
Is AP Skarner viable? Should he build magic pen considering how much magic damage he's doing?
No, it isn't viable. His AP ratios were nerfed too hard. You can still kind of make it work though, and if you do choose to use it, I wouldn't build magic penetration except potentially as your final item. You have effects that won't benefit from magic penetration such as boosting your shield, and you still need some defensive AP items, so penetration is a lower priority.