Bdtyktl

I have a problem that I believe is NP-hard. If I reduce a NP-complete problem to it in exponential time (and not polynomial time), does that prove the problem is NP-hard?

Hardness/completeness is always defined with respect to a specific kind of reduction.

Using exponential-time reductions doens't work at all, because that means your reduction is even more powerful than the thing you're reducing to.^*
Every language except $emptyset$ and $Sigma^*$ is NP-hard under exponential-time reductions. To see this, let $L$ be any language except $emptyset$ and $Sigma^*$, and let $X$ be in NP. We can reduce $X$ to $L$ as follows. Fix two strings $w_{mathrm{yes}}in L$ and $w_{mathrm{no}}notin L$. Now, given a string $x$, we can determine in exponential time if $xin X$. If it is, the reduction maps $xmapsto w_{mathrm{yes}}$; otherwise, it maps $xmapsto w_{mathrm{no}}$.

^* OK, technically, we don't know that EXP is more powerful than NP, but it's certainly at least as powerful, and probably more powerful.