Are accepting states and unaccepting states distinguishable? Is it true that for all DFA that if $A$ is an accepting state and $B$ is not, that $B$ is always distinguishable from $A$.

I don't think it's true, just because I think I can create a DFA.

$B$ reads b -> $A$

$A$ reads b -> $A$

If you start at $B$ while reading a b and at $A$ you read a b you end up in the accepting state $A$. Now is there an algorithm to see if my construction is valid?
This depends on your definition on distinguishable. Let's assume the following definition: $A,B$ are distinguishable if for some word $x$, if the automaton starts at $A$ and reads $x$ then it accepts, while if it starts at $B$ and reads $x$ then it rejects; or vice versa.

Under than definition, any accepting state is distinguishable from any non-accepting state: just take $x$ to be the empty string.