"I thought that ethoxide ion (a strong nucleophile) would cause the compound to undergo an SN2 reaction because ethoxide isn't bulky. Also, why would the compound want to have the unstable double bond at the end formed by an elimination reaction??"
It depends on whether the substrate is primary or secondary. All else held equal, a secondary alkyl halide is more likely to show elimination than a primary one is.
"I found a separate commentary in the answer key that said 'we know it's E2 because a strong base was used." To clarify, using a strong base doesn't mean that it's more likely to be E2 rather than Sn2, correct?'
My interpretation of this passage is that the answer key is that this favors the E2 mechanism over the E1 mechanism; however, that's just my inference, which may not have been what the author intended.
With respect to the question of strong bases and E2 versus SN2, I used to think that bulky bases favored elimination more than small bases. However, I looked up some data last year that made me think that the story was a bit more complex than that (I don't have the data handy at the moment). I do think that it's fair to say that there are multiple variables (temperature, solvent, base strength versus nucleophilic strength, perhaps some more) that must all be considered.