Problems that can be used to show polynomial-time hardness results
Yes, the best known algorithm for $k$-SUM runs in $O(n^{\lceil k/2 \rceil})$ time, so it's very possible that you could argue some $n^7$ problem is difficult, because if it's in $n^{6.99}$ then you can solve $14$-SUM faster. 

Note the $k$-SUM problem gets "easier" as $k$ increases: given an improved algorithm for $k$-SUM, it's fairly easy to get an improved algorithm for $2k$-SUM: take all $O(n^2)$ pairs of $n$ numbers in your given $2k$-SUM instance, replacing each pair with the sum of the two, and look for a sum of $k$ numbers out of those that equal $0$. Then, an $O(n^{k/2-\varepsilon})$ algorithm for $k$-SUM implies an $O(n^{k-2\varepsilon})$ algorithm for $2k$-SUM. Put another way, a tight lower bound for $2k$-SUM is a stronger assumption than a tight lower bound for $k$-SUM.

Another candidate for a hard problem is $k$-Clique. See my $O(\log n)$-Clique answer for more on that. If you can show (for example) that a better algorithm for your problem implies an $O(n^2)$ algorithm for $3$-clique, then a super-breakthrough would be required to improve on your algorithm. Parameterized complexity gives many examples of other problems like this: $k$-Clique is hard for the class $W\[1\]$, and $k$-SUM is hard for $W\[2\]$.

Let me warn you that although problems like this are very convenient to work with, problems like $3$-SUM are not among the "hardest" in $TIME[n^2]$, e.g., it is very unlikely that every problem in $TIME[n^2]$ can actually be linear-time reduced to $3$-SUM. This is because $3$-SUM can be solved with $O(\log n)$ bits of nondeterminism in linear time, so if everything in quadratic time can be reduced to $3$-SUM, then $P \neq NP$ and other fantastic consequences results. More on this point can be found in the article "How hard are $n^2$-hard problems?" (At some point, "3SUM-hard" was called "$n^2$-hard"; this SIGACT article rightly complained about that name.)