Just got some wheels today and put them on my bike. First thing that I thought was weird was that I had to use quite a bit of force to pull the fork wide enough to fit onto the wheel. Second, when the wheel is all the way into the dropouts, it's skewed WAY to the right.
I know this isn't right. I can put the wheel in the center of the fork like it's supposed to be, however, it won't be in one dropout all the way, if that makes sense. I think the problem is that one of the dropouts is bent. Or, one dropout is "deeper" than the other. Ideas?

Something is definitely wrong with your wheel.

Do the old wheels fit the same way? If all the wheels fit the same, I'd blame the fork, if just one wheel doesn't fit right, I'd blame the wheel.
A good shop will have fork alignment jigs and will be able to straighten it if it's just bent. (assuming the fork is steel)

1. Loosen the track nuts, does the wheel go center?
2. Flip the wheel, is it now off to the other side?
3. Put a different wheel in (known to be dished to center), does it go to the side?
4. With loose nuts, push hard to see if wheel seats in fork ends.

By bending the fork, you probably bent one of the forks out of alignment ? Easiest way I know to check it, install the fork like you have, take the wheel out and line up the top of the fork to be a perfect 90* orientation with the rest of the frame longitudinally (probably like your pic shows you've already done ?) on a tile floor. Make sure you have the fork dropouts where 2 tiles meet. If one dropout is more forward than the other, the bending you did has misaligned your fork & steering. And you'll know it when you put the wheel in it and ride it. Your bike is turning either left or right on it's own.

If the steering is out of alignment. This initially & will prematurely wear your front tire more, as you'll wind up correcting the direction of the bike as you ride it. It may eventually & prematurely wear your bearings out in the front wheel too. This is the equivalent of a car that has misaligned steering for a bike. Same issues hold with a bent frame too. To go perfectly straight the frame (north to south) needs to be in alignment from the steerer tube to the seatpost tube. Behind the seat tube, that also needs to be in perfect alignment, otherwise the bike frame will try to turn itself on it's own too. The steering tube is a 90* (east to west) orientation with the rest of the frame to go perfectly in a straight line.

You can also tell just by walking the bike down a flat surfaced hallway or sidewalk. Align the top of the steering fork in the 90* orientation with the frame and using you hand on the seat only, start to walk the bike forward and down the middle patch of both tires. If it goes in a straight line, everything is right, if it starts to turn or veer to either the left or right, the fork is bent, seeing how that's what you bent and assuming the rest of the frame was welded together in alignment.

Put in an old 27 inch wheel: it's so far to the right it rubs the side of the fork. A magnet sticks to it, so I'm assuming it's steel.

I didn't really BEND the fork, rather, I "stretched" it a bit to fit the wheel in. I have to repeat the same process every time I put a wheel in, so I haven't permanently bent it.

1. Nope.
2. Nope.
3. Yep.
4. It definitely seats all the way, it just goes crooked when this is done.

Take it to a good shop in Davis or email the guy at Innerlight Cycles (local Davis framebuilder) to see about alignment. Or go down to Bike Forth and find a new fork for cheap.

Bent fork.

Find a shop with an alignment tool/apparatus and someone who knows how to use it or get a new fork.... or just ride it.