"Logical implication" is a potentially misleading term; it may mean the propositional connective often called Conditional.
In this case : YES, having two statements $P,Q$ we can always produce the "complex" statement $P → Q$, that reads :
"if $P$, then $Q$".
A different (but related) case is when we use "implies" to mean Logical consequence :
a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements.
In this case we use the symbol : $Γ \vDash \varphi$, that reads : "statement $\varphi$ logically follows from the set $Γ$ of statements".
Statements are $2=2$ (which is True) and $2=3$ (which is False). To evaluate the truth value of a "complex" statement (like $P → Q$) we have to start from statements having a precise truth value.
$x=2$ is not a statement : it is a formula with a variable and its truth value depends on the value assigned to variable $x$.
A different case is when we have quantifiers, like e.g. $∀x(x=2 → x>1)$. In this case there are no more free variables and the formula is a statement : if we read it as a formula about natural numbers, it has a precise truth value : it is True.
Regarding your examples, we have that $\forall x (x=2 \to x^2 < 6)$ is always True (as you say) when red as an arithmetical statement, while $\forall x \forall y (x=2 \to y=5)$ is not.
how can the truthfulness of the statement $P \to Q$ be variable, depending on context?
$P \to Q$ is a formula of propositional calculus.
Formulas of propositional calculus are Truth functions meaning that :
a compound statement is constructed by one or two statements connected by a logical connective; if the truth value of the compound statement is determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and the logical connective is said to be truth functional.
This means exactly that, in order to evaluate the truthfulness of the statement $P \to Q$, we have to specify a "context", i.e. a truth assignment, that is a function that maps propositional variables to True or False.
In this way, given a "context" (a truth assignment), then YES : the truth value of a (truth-functional) compound statement, like e.g. the conditional $P \to Q$, is always determinable from the given statements $P$ and $Q$.
Best Answer
I think you are mistakenly determining actual truth values for each of the pairs.
In the first question, it isn't important that normally every multiple of three is not odd. I think the real point is that IF it were the case that every multiple of three were odd, then it would also be the case that some multiples of three are odd. Thus the two statements are neither contrary nor contradictory.
In the second question, the two statements are contradictory like you said (and more obviously so because we have immediate examples available to us of models of geometry in which one is true but not the other).
The third pair of statements are also contradictory because they are exact negations of each other.