Expressiveness of Büchi vs CTL(*)
I'm not answering the full question but only part of it (I have no interest in branching time).

Your definition of $\mathit{even}$ would better read $\mathit{even}(p) \equiv \exists q.(q \land \Box ( q \leftrightarrow \mathsf{X} \lnot q ) \land \Box ( q \rightarrow p ))$.  You are introducing $q$ only to remember if you are on an odd or even position, but this $q$ information is not on your system so it should not be a free variable of your formula (otherwise your system and your formula are defined on different alphabets).  Such a formula is an Existentially-Quantified LTL formula (EQLTL for short).  

EQLTL formula can only have $\exists$ quantifiers at the top-level.  And they are as expressive as Büchi automata.  For instance to translate
the EQLTL formula $\exists q.(q \land \Box ( q \leftrightarrow \mathsf{X} \lnot q ) \land \Box ( q \rightarrow p ))$ as a Büchi automaton, you would simply translate the LTL formula $q \land \Box ( q \leftrightarrow \mathsf{X} \lnot q ) \land \Box ( q \rightarrow p )$, and then erase the $q$ variable from everywhere in the automaton.   Conversely any Büchi automaton can be converted into an EQLTL formula easily: declare an existential variables for all the states $\exists s_1.\exists s_2\ldots$, and use these to encode transitions and acceptance states (e.g. $\exists s_1.\exists s_2.(s_1\land\Box(s_1\rightarrow a\land \mathsf{X}s_2)\land\Box(s_2)\rightarrow b\land\mathsf{X}(s_1))\land\Box\Diamond s_2\land\Box(\bigwedge_i (s_i\rightarrow\bigwedge_{j\ne i}\neg s_j)))$ if $s_1$ can go to $s_2$ with $a$, $s_2$ can go to $s_1$ with $b$, and $s_2$ is Büchi-accepting; the last conjunction of implications make sure you cannot be in two states at once).  You might want to read Stutter-Invariant Languages, ω-Automata, and Temporal-Logic on this subject.

Basically the $\exists q$ is used here like some second-order quantifier (there exist a set of states where $q$ hold and such that...).  EQLTL and Büchi automata are equalivent to S1S, and LTL is equalivent to F1S.  $even$ is the typical S1S formula that cannot be expressed by F1S.

QPTL, which you mentioned, adds $\exists$, $\forall$ (not only at the top-level), and past operators.  It contains EQLTL but is not more expressive (since QPTL can be converted to Büchi automata and Büchi automata can be converted to EQLTL).

CTL is about branching time, so it's a different world.  A CTL formula like $\mathsf{EF}\mathsf{AG}p$ which is reasoning on the tree of all possible futures have now equivalent in LTL which is reasoning on all possible scenarios.