Pushdown Automata - can you have multiple transition functions options between 2 states?
Simple answer: yes, that is allowed. In the example you give, it's simple--depending on the input symbol and the stack contents, there's one and only one choice for the action of the machine in state $q_0$. It's even possible to have several actions as consequence to a single (input, stack top) pair, like this,

-- input 0 and stack top 1, push 0 and go to state $q_2$

-- input 0 and stack top 1, push 1 and go to state $q_3$

Seems wierd, I know but such machines are called nondeterministic pushdown automata and are quite useful; look them up.