This was very important before computers in problems where you had to do something else after computing an answer.
One simple example is the following: When you calculate the angle between two vectors, often you get a fraction containing roots. In order to recognize the angle, whenever when possible, it is good to have a standard form for these fractions [side note, I saw often students not being able to find the angle $\theta$ so that $\cos(\theta)=\frac{1}{\sqrt{2}}$]. The simplest way to define a standard form is by making the denominator or numerator integer.
If you wonder why the denominator is the choice, it is the natural choice: As I said often you need to make computations with fractions. What is easier to add:
$$\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{6}+\sqrt{3}} \, \mbox{ or }\, \frac{\sqrt{3}}{3}+\frac{\sqrt{6}-\sqrt{3}}{3} \,?$$
Note that bringing fractions to the same denominator is usually easier if the denominator is an integer. And keep in mind that in many problems you start with quantities which need to be replaced by fractions in standard form [for example in trigonometry, problems are set in terms of $\cos(\theta)$ where $\theta$ is some angle].
But at the end of the day, it is just a convention. And while you think that $\frac{1}{\sqrt{2}}$ looks simpler, and you are right, the key with conventions is that they need to be consistent for the cases where you need recognition. The one which looks simpler is often relative...
It's useful in everyday life. Most people will only ever come across fractions when dividing objects between groups, and then it is useful. For example, if you have to split 16 things between 5 people you do $\frac{16}{5} = 3\frac{1}{5}$ and you know you'll have three each plus one left over. This is much easier (and practically more useful) than actually doing the division to find the decimal expansion of $\frac{16}{5} = 3.2$.
Best Answer
Firstly, $\frac{p}{0}$ never becomes $\infty$, it is simply undefined.
Now, a fraction $\frac{p}{q}$, with $p,q$ integers and $q\ne 0$, is introduced in order to solve a problem. The problem is to solve the equation $q\cdot x = p$. This solution, even though described with integers alone, does not always have a solution in the integers. So, we extend the integers by introducing solutions to all such equations as long as it makes sense. An example where it does not make sense is the equation $0\cdot x = p$, where $p\ne 0$, since we wish to retain the fact that $0\cdot x=0$. There are some subtleties here, solutions to different equation may lead to essentially that same fraction. This is solved by the familiar equations of the form $\frac{1}{2}=\frac{2}{4}$ and so on.
Now, fractions with negative denominators simply are solutions for equations of the form $-7\cdot x = 2$ or $-5x=17$. There is no reason to exclude such solutions. In fact, by adjoining all solutions to all equations of the form $qx=p$, where $p,q$ are integers and $q\ne 0$ (and properly identifying fractions that only appear to be different in form) one obtains the field of the rational numbers.