Understanding reductions: Would a polynomial time algorithm for one NP-complete problem mean a polynomial time algorithm for all NP-complete problems?
But why would I assume the converse holds, that I can solve arbitrary instances of 3-coloring provided an oracle for 3SAT?


3-SAT is $NP$-complete, by the Cook-Levin theorem. This means that any problem in $NP$ (such as 3-coloring) can be solved in polynomial time given an oracle for 3-SAT.


  shouldn't we need to show $Q \leq_p P$ AND $P \leq_p Q$, where $Q$ is some known NP-complete problem, to prove that some problem $P$ is NP-complete?


Yes, but you essentially get the $P \leq_p Q$ for free because $P$ is in $NP$ and $Q$ is $NP$-complete.