This seems to imply that there are other relations for which this restriction need not hold. My first question is: what are some examples of such relations? And second, if there are other relations out there, then why are functions so important? Why are they what is primarily studied in algebra and calculus?

A common example of an important equation that isn't a function would be the formula for a circle or something like y^2 + 3x = 6, since you cannot solve for a unique y.

I think the importance of function can turn into a very long and large post. I think it should suffice to say that functions have a lot of very handy and interesting properties. I'm not a historian, so I can't tell you exactly why functions became the main focus in a lot of mathematics. I can say that once you begin to really study functions you can say a lot about them and find out a lot more about other mathematical ideas, and maybe, if there is no other reason, that's why we choose to focus on them so much.

Important examples of relations are of course the equality relation =, the not-equality relation ≠, the inequalities <,>,≤,≥. We also have the relation "is parallel to" or "is perpendicular to" between straight lines. Most of these relations are not functions and are very important in mathematics.

One of the most important reasons for studying functions is, I guess, that they arise naturally in physical applications. For example, let's say that I drive my car on the highway. When I start driving, I start my stopwatch. So at time 0, I am at my starting position. After 10 minutes, I might be 5 miles from my starting position. After 1 hour, I might be 20 miles from my starting position.
The key is that after a certain time t, I am a certain number of miles from the start. So I get a function in a very natural manner: at a certain time I can't be 10 and 20 miles from my starting position, this is impossible. So for each time, I can associate a unique distance that I am from the start. So I have a function.
As such, functions arise in a very natural manner. Furthermore, functions are really easy to handle.

Sure, for example: define xRy if x=y^{2}. Then 1R1 and 1R(-1), so 1 is in relation with exactly two other numbers. In general, if x≥0, then [itex]xR\sqrt{x}[/itex] and [itex]xR(-\sqrt{x})[/itex].