NP-hard in what parameter? I often read statements such as "the run-time of algorithm X is polynomial in m and n". Indeed, when a problem has two parameters, it is possible that the run-time of an algorithm is polynomial in one parameter but not the other one, so if this is not the case, it is important to emphasize that it is polynomial in both parameters.

However, I have never read a statement such as "the problem is NP-hard in m and n". A problem is always claimed to be NP-hard, period. My first question is: why? Apparently the same rationale is true here too: a problem may be NP-hard in one parameter, but at the same time, it may have an algorithm with run-time polynomial in the other parameter. 

My second question: what is an accurate description of a problem with two parameters: $m$ and $n$, that (1) for a fixed $m$, can be solved in time polynomial in $n$, (2) for a fixed $n$, it is NP-hard (with respect to $m$)?
A problem is always claimed to be NP-hard, period.


Indeed, a problem's definition already contains a specification of its parameters. (See the entries in Richard Karp's seminal collection of NP-complete problems for several examples.) Usually, there is no need to make explicit reference to the parameters per se, as they are "automatically scaled" by the input length; for your two parameters $m$ and $n$, for instance, the input could simply be encoded as $0^m1^n$. Hence, for abstract problem instances, it usually suffices to refer strictly to the input length, which is indeed how time complexity is defined (i.e., as the number of computation steps as a function of the input length).

Studying how different requirements on the parameters impact the problem's complexity is the object of study of parameterized complexity theory. For example, the set of SAT formulas with $k$ many variables, $k$ being a fixed parameter (i.e., constant), is decidable in polynomial time (in the formula's length); in parameterized complexity terms, this means SAT is fixed-parameter tractable. It is only as the number of variables is also allowed to scale (in the input length) that SAT becomes NP-hard.