Propositional Logic - Associative Property question $p \land \lnot q \lor q \land \lnot r \lor \lnot p \lor r $
$\equiv$$(p \lor \lnot p) \land (\lnot q \lor q) \land (\lnot r \lor r)$
Is this move "legal"? Or can you only apply the associative property on like operators? 
 A: Associativity applies only when the connectives involved are exclusively $\land$ or exclusively $\lor$:
$$p \land q \land r \equiv (p \land q)\land r \equiv p \land (q\land r)$$
$$p \lor q \lor r \equiv (p \lor q)\lor r \equiv p \lor (q\lor r)$$
Because of associativity of $\lor$ and $\land$, parentheses are not necessary to define expressions like those above.
Your statement, however:
$$p \land \lnot q \lor q \land \lnot r \lor \lnot p \lor r \tag{given}$$
has mixed connectives, and so associativity does not apply across all possible groupings. 
Please note: as stated, your (given) expression is not well-defined without parentheses. That is, without parentheses, it is ambiguous; it can be read any number of ways, most of which are not equivalent. Does it mean connect from left to right?:
$$(((((p\land \lnot q) \lor q) \land) \lnot r)\lor\lnot p) \lor r\;?\tag{1}$$
Or does it mean this?
$(p \land \lnot q) \lor (q \land \lnot r) \lor (\lnot p \lor r)\;?\tag{2}$
or any number of other possible ways of grouping with parentheses?

In general, when you have an expression like $(2)$ above, you need to apply the Distributive Laws to distribute over another connective:
For example $$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$
$$p \lor (q\land r) \equiv (p \lor q) \land (p\lor r)$$
A: No, mixed expressions like this are not associative; instead, they obey distributive laws:
$$
(a\wedge b)\vee c \equiv (a\vee c) \wedge (b \vee c)
$$
and
$$
a \wedge (b\vee c) \equiv(a\wedge b) \vee (a \wedge c).
$$
