Does NP-completeness/hardness have to be constructive? Is there any $L\in {\bf NP}$ with the following properties:


It is known that $L\in {\bf P}$ implies ${\bf P}={\bf NP}$.
There is no (known) polynomial time Turing reduction of $SAT$ (or some other ${\bf NP}$-complete problem) to $L$.


In other words, if a polynomial time algorithm for $L$ implies the collapse of ${\bf NP}$ into ${\bf P}$, then  is it necessary that this "general hardness" of $L$ for ${\bf NP}$ must be somehow $constructive$, in the sense that, say, $SAT$ must be reducible to $L$ via some specific reduction?
Yes, there are such sets, take any $\mathsf{NP}$-intermediate set (any set that is provably $\mathsf{NP}$-intermediate assuming $\mathsf{P}\neq\mathsf{NP}$), e.g. construct one from SAT using Ladner's theorem.

Note that your $L$ needs to considered an $\mathsf{NP}$-intermediate problem, since it is in $\mathsf{NP}$ but not complete for it. Note also that you are assuming that $\mathsf{P}\neq\mathsf{NP}$ otherwise there is no such $L$ as every non-trivial problem would be complete for $\mathsf{NP}$ if $\mathsf{NP}=\mathsf{P}$. Additionally the conditions that you have given does not imply to completeness so the question in the first part is not the same as the question about constructiveness of completeness.



Regarding the question in the title, i.e. "does $\mathsf{NP}$-hardness have to be constructive?". 

The answer depends on what we mean by "constructive". Classically, a decision problem $A$ is defined to be $\mathsf{NP}$-hard iff

$$\forall B\in \mathsf{NP} \ B \leq^\mathsf{P}_m A$$

which means

$$\forall B\in \mathsf{NP} \ \exists f\in\mathsf{FP} \ \forall x\in\{0,1\}^* \ (x\in B \leftrightarrow f(x)\in A)$$

And by Cook's theorem this is equivalent to

$$ SAT \leq^\mathsf{P}_m A$$

which means 

$$\exists f\in\mathsf{FP} \ \forall x\in\{0,1\}^* \ (x\in SAT \leftrightarrow f(x)\in A)$$

How can we make this definition constructive? It is already seems very constructive to me.
I guess what you want to ask is if we can prove this for some $A$ without knowing what is $f$ explicitly. I don't remember seeing any such hardness proof. 

Classically even when we don't have a specific function there is a function, saying that it is impossible that no function is a reduction is equivalent to saying that some function is a reduction. To talk about constructiveness we need to be more considerate. For example we can talk about statements which are provable classically but not constructively (e.g. intuitionism where different state of mathematical knowledge makes sense, Google for "ideal mathematician" or check this).

Intuitively it seems plausible to me that we can prove such a statement using a proof by contradiction and without giving any explicit reduction function. But it won't mean that there is no constructive proof of the statement. To be say more that no constructive proof exists we have to be more specific: proofs in which theory/system? what do we mean by a constructive proof?