Lets say I have some predicate logic formula in Boolean algebra and I have to show whether the formula is logically valid. Can I show it by showing that the formula follows from the axioms of the Boolean algebra? Or can I only prove it for example by creating a semantic tree for negation of that formula and checking whether the formula is a contradiction and therefore not negated formula a logically valid formula?
Also, is it always the case that every axiom is logically valid? Why?
No, the axioms of Boolean Algebra are not logically valid.
Logically valid means:
If they were, also the theorems (laws) of BA would be, and there are obvious laws of BA that are not true in every interpretation, like e.g.: $x+x=x$.
And yes, in order to prove that the above formula is a theorem of Boolean Algebra (i.e. that it follows form the axioms), you can apply the semantic tree proof procedure to the set $\mathsf {BA}$ of axioms and the negation of the law to be proved.