Is there a cusped (non-compact) orientable hyperbolic 3-manifold $M$ which has Heegaard genus $g$, and has a finite-sheeted cover with Heegaard genus $<g$? Moreover, if the cover is index $n$, then it should have fewer than $kn$ cusps, where $k$ is the number of cusps of $M$ (so it does not induce the trivial $n$-fold cover of each cusp component).

**Remarks:** I believe this is an open question. To clarify, by Heegaard genus $g$ of a cusped hyperbolic manifold $M$, I mean that $M$ is the interior of a compact manifold $N$ with torus boundary components, so that $N$ has Heegaard genus $g$, i.e. there is a surface of genus $g$ in $N$ splitting it into two compression bodies. For closed manifolds, there are examples of Alan Reid. One may modify these examples to get cusped examples which are the trivial cover of the cusps. For more discussion in the closed case, see this paper of Peter Shalen.

I would also be happy with a 1-cusped manifold of Heegaard genus 3, and which has a $k$-fold cover with a maximal embedded horocusp of volume $\leq \pi$ and $<k$ cusps (e.g. an irregular 3-fold cover with 2 cusps). (although Nathan's answer answers the original version of this part of the question, I realized I hadn't given the correct formulation).