This post is not about counterintuitive tautologies, but about conterintuiitive contingent formulae.
Easy examples of formulae that ( probably) are counterintuitively non-antilogical are : $ \neg A \rightarrow A$ and $ A \rightarrow \neg A$ . Also : $ (A \rightarrow B) \land (A\rightarrow \neg B) $.
Based on your experience ( maybe in teaching logic) are there other examples you can point to?
What does it mean to say that it is “counterintuitive” that a given logical formula is contingent?
Take, say, your third example $(A\to B) \land (A \to \neg B)$. Anyone who knows the truth-table for $\to$, knows this comes out true when $A$ is false. So it is not even initially surprising that the formula has true instances and is contingent -- let alone “counterintuitive” (and I think “counterintuitive” is usually taken to mean something stronger than “initially surprising”).
If there is anything counterintutive in the vicinity, it isn't that the logical formula “$(A\to B) \land (A \to \neg B)$” on the standard interpretation of the connectives is contingent.
Perhaps you meant instead that it is counterintuitive that there are instances of the schema “Both if $A$ then $B$, and also if $A$ then it is not the case that $B$” (involving now the ordinary language conditional) which are true. But that doesn't seem obviously right either. Anyone who has met a reductio argument for showing that $A$ implies a contradiction can massage that into a proof that if $A$ then $B$ and a proof that if $A$ then not-$B$ -- so knows in this case that both if $A$ then $B$ and also if $A$ then it is not the case that $B$.
If there is something counterintutive roughly in the vicinity -- picking up on the fact that your supposed examples all use the conditional -- it is the straight-out claim (in some logic books, at any rate) that the truth-functional arrow is an adequate rendition of ordinary-language “if”. But if(!) that is what is at the back of your question, then I don't think the issues here are best brought into focus by raising questions about the contingency of various logical formulas.