Do most mathematicians know most topics in mathematics? How many topics outside of his or her specialization is an average mathematician familiar with? 

For example, does an average group theorist know enough of partial differential equations to pass a test in a graduate-level PDE course? 

Also, what are the "must-know" topics for any aspiring mathematician? Why?

As a graduate student, should I focus more on breadth (choosing a wide range of classes that are relatively pair-wise unrelated, e.g., group theory and PDEs) or depth (e.g., measure theory and functional analysis)?
Your question is philosophical rather than mathematical. 

A colleague of mine told me the following metaphor / illustration once when I was a bachelor student and he did his PhD. And since now some years have passed I can relate.

It is hard to write it. Think about drawing a huge circle in the air, zooming in, and then drawing a huge circle again.

This is all knowledge:

[--------------------------------------------]


All knowledge contains a lot, and math is only a tiny part in it - marked with the cross:

[---------------------------------------x----]
                                        |
Zooming in:
[xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx]


Mathematical research is divided into many topics. Algebra, number theory, and many others, but also numerical mathematics. That is this tiny part here:

[xxxxxxxxxxxxxxxxxxxoxxxxxxxxxxxxxxxxxxxxxxxx]
                    |                    
Zooming in:
[oooooooooooooooooooooooooooooooooooooooooooo]


Numerical Math is divided into several topics as well, like ODE numerics, optimisation etc. And one of them is FEM-Theory for PDEs.

[oooooooooooooooooooρoooooooooooooooooooooooo]
                    |                    


And that is the part of knowledge, where I feel comfortable saying "I know a bit more than most other people in the world". 
Now after some years, I would extend that illustration one more step: My knowledge in that part rather looks like

[   ρ    ρρ  ρ         ρ   ρ          ρ     ρ]


I still only know "a bit" about it, most of it I don't know, and most of what I had learned is already forgotten.

(Actually FEM-Theory is still a huge topic, that contains e.g. different kinds of PDEs [elliptic, parabolic, hyperbolic, other]. So you could do the "zooming" several times more.)



Another small wisdom is: 
Someone who finished school thinks he knows everything. Once he gained his masters degree, he knows that he knows nothing. And after the PhD he knows that  everyone around him knows nothing as well. 



Asking about your focus: IMO use the first few years to explore topics in math to find out what you like. Then go deeper - if you found what you like. 

Are there "must know" topics? There are basics that you learn in the first few terms. Without them it is hard to "speak" and "do" math. You will learn the tools that you need to dig deeper. After that feel free to enjoy math :) 
If your research focus is for example on PDE numerics (as mine is) but you also like pure math - go ahead and take a lecture. Will it help you? Maybe, maybe not. But for sure you had fun gaining knowledge, and that is what counts. 

Don't think too much about what lectures to attend. Everything will turn out all right. I think most mathematicians will agree with that statement.
Certainly not. For example, the great mathematician Grothendieck was insufficiently well acquainted with arithmetic to recognize the integer $57$ as a non-prime. The many accounts of this story can be accessed by an internet search for the key terms; say, look for grothendieck prime 57.

An earlier example is given by Ian Stewart on page 72 of his book Professor Stewart's Cabinet of Mathematical Curiosities, which I quote verbatim without any endorsement as to the degree of its truth:

Ernst Kummer was a German Algebraist, who did some of the best work on Fermat's Last Theorem before the modern era. However, he was poor at arithmetic, so he always asked his students to do the calculations for him. On one occasion he needed to work out $9\times7$. "Umm ... nine times seven is ... nine times ... seven ... is ..."

$\qquad$"Sixty-one," suggested one student. Kummer wrote this on the blackboard.

$\qquad$"No, Professor! It should be sixty-seven!" said another.

$\qquad$"Come, come, gentlemen," said Kummer. "It can't be both. It must be one or the other."