In several comments, it was mentioned "just make the units of $N_0$ whatever they need to be, without concern as to why. I thought I might add to that.
Units are a funny thing because in many cases, they serve no deeper purpose than to ensure you did the math correctly. Take Hooke's law as a classic example. Hooke's law is most accurately described in mathematics as $F\propto x$, where F is the force exerted by a spring is proportional to x, which is the amount the spring is extended or compressed. However, this very pure form is inconvenient for doing math. Instead we rewrite that equation as $F=kX$, where $k$ is a measure of the stiffness of the spring. The two expressions convey the same information, but one uses proportionality while one uses equality and a constant multiplier.
When converting proportionality to equality, we have to deal with the fact that proportionality and equality differ in how they deal with units. Proportionality is blind to units. If you do the math, the units always end up canceling out. Equality, however, does require the units to line up. To make them line up we assign a unit to the constant which makes the equation work.
Sometimes we find that these assigned units have a deeper meaning, other times they do not.
So in your case, you write $n(r)=N_0r^2$, but in words you chose to write "...the number density varies with radius," which is a proportionality phrasing $n(r)\propto r^2$. $N_0$ only appears as a side effect of turning that proportionality into an equality, and thus the units of $N_0$ are safely assigned to "whatever they need to be to make the units work." This also points out the answer to your second question: if you used a distribution where $n(r)\not \propto r^2$, then you would have a different equation when you converted it to equality. Thus, you would need to make up a different unit.