I'll answer what I can of your questions (in blue). 
First, I live alone in a two-bedroom apartment. Will a Yorkie be content in a small space? I plan on walks and frequent trips out of the home, but I worry that a dog would be MUCH happier with a yard ...
I would say, of course, a yorkie would be much happier with a yard. Any dog enjoys their own little yard to run and play in and it does become THEIR yard, if you get my meaning. 
But since you are planning to move in a year and the yorkie you are getting is a small yorkie and not a big yorkie then you will be fine.
Also, I know Yorkies can be difficult to potty train. I plan on crate training and using pee pads (since I'm in an apartment.) Are they really that much more difficult than other breeds? I potty trained my mother's miniature American Eskimo with little problems, just a few accidents.
I don't find them any more difficult than other breeds. I train ours to go outside. I (myself) believe this rumour of their difficulty in housebreaking came about with the use of pee pads instead of training to the outdoors.
Also, I was thinking of getting a playpen. Would this work as well as a crate? Will a playpen hold an older Yorkie? Will baby gates hold a Yorkie (my apartment has stairs). If I get a playpen, do I really need a crate?
It depends on the size of the yorkie. They come in all sizes.
Many of the neighbors in this community have dogs as well, some pretty large. How likely is a Yorkie to start a fight? I know they can be a bit rambunctious ...
Are you talking about a little yorkie on a leash or do you plan to allow it to run loose? If it is on a leash, then there should be no problem.
In a year, I will be moving about 30 minutes away to a house with my friend. The house will be fenced. Do you think it will be difficult to go from pee pad training to having a yard as well? Do you think a move (albeit small) will upset my Yorkie?
Yes, I do.
What do ya'll suggest I do for socialization?
Allowing your puppy to meet people on a regular basis, puppy play dates...an introduction to car rides (slowly at first); just a continued introduction to different surroundings and situations, but not too much at a time. A timid puppy can be easily frightened. Training classes.
While I'm at work, would puppy day care be a good idea or should I leave her home alone for about 6 hours?
If you can afford puppy day care, then that would be a plus. Six hours is not terribly long. Is that five days a week or more?