Common insights into hypothetical complexity of graph problems
Here are two references for the second part of your question.

The paper [1] addresses certain types of colorability of sparse graphs with given girth $g$. For every fixed $g$, they show that the associated decision problem is either trivial (every graph in the class has a coloring) or NP-complete. But determining which is the threshold value of $g$ remains a difficult open problem!
Edit: One of the considered problems is related to Jaeger's conjecture, that every planar graph of girth $4k$ admits a homomorphism to $C_{2k+1}$. It is shown in [1] that any counterexample directly provides a hardness proof. (A similar conjecture by Klostermeyer and Zhang exists for the odd-girth.) For the other problems considered in [1], there is no official conjecture, but for any guess about the correct threshold value $g$ that one can make, if proved false by a counterexample, the latter directly implies a corresponding hardness proof.

In the introduction of the above cited paper is also mentioned the following interesting result about SAT [2]. It is proved there that for every $k$, there exists a function $f(k)$ such that $(k,f(k))$-SAT (i.e. $k$-SAT where each variable occurs $f(k)$ times) is trivial, but $(k,f(k)+1)$-SAT is NP-complete. (The precise value of $f(k)$ seems unknown, although some estimate is given.)

[1] L. Esperet, M. Montassier, P. Ochem and A. Pinlou. A complexity dichotomy for the coloring of sparse graphs. Journal of Graph Theory 73:85-102, 2012. link + PDF on an author's website

[2] J. Kratochvil, P. Savicky and Zs. Tuza. One more occurrence of variables makes satisfiability jump from trivial to NP-complete. SIAM Journal on Computing 22:203-210, 1993. link