How do I normalize a push down automaton?
What do you call a normalized PDA? There are many ways to specify
that. Also, a PDA is defined by its transition fonction. The diagram
is just a convenient graphical way to do that, as long as it is
precise and easily read.

For the example you give, a transition is not represented by a
directed edge, but by a directed edge together with a label. If you
have an edge with two labels, as in your loop on $q_1$, that
represents two transitions. Trying to do it with two separate loop
edges would only make your diagram less readable.

Now, it is possible that your instructor has defined "normalized
diagrams" in some way, and you should try to follow his wishes.  But I
do not remember of any such normalization of PDA diagrams with real
technical importance.

Yes it is OK to have multiple transitions from one state to
another. You could avoid it by introducing new intermediate states,
but that would serve absolutely no purpose other than making your life
miserable.