I actually find this a strange cause to go to war: did southerners in general really feel this strongly about the issue of slavery?
Pretty much. I'll talk about this more below.
I suppose the fact most voters in the South were slave owners had something to do with it
I'm pretty sure that's not true. Looking at wikipedia's article on the election (too lazy to get a real source), approximately 840,000 people cast ballots in the states that seceded (including West Virginia but not including South Carolina). As an aside, that's about equal to the number of votes Breckenridge, the Southern Democrat, got, which totals about 18% of the vote. Anyway, discounting slaves, the population of the Confederate states was about 5.5 million. Halved (assuming it wasn't already) to account for women not being able to vote, that would mean about thirty percent of the male white population of the South voted in 1860.
By comparison, the "North" had a population of roughly 22 million. So, even if every vote in 1860 were cast in those states, only about 42% of the male population voted. This, it should be noted, does not take into account slaves in the border states, which I assume were counted for that tally. And, of course, this makes no effort to account for things like people not being old enough to vote.
So, even if just half of that 840,000 were slave owners, there would only be an average 6 slaves per owner, which doesn't strike me as particularly viable, given the use of slaves as a labor force on plantations.
Still, I don't think it's a coincidence that the one state that still let the legislature pick its electors was also the first to secede and always the most insanely pro-slavery of them all.
when they perceived their livelihood to be threatened (the declaration even mentions that they fear for their economy). Were the southern armies generally composed of eligible voters, though? How was their morale?
Southerners who fought seem to have generally perceived themselves as defending their homeland from aggressors (I remember some quotes in a thread here awhile back warning of the North's "abolitionist host" coming to rape and pillage). Slavery was inextricably wedded to Southern society, so any threat to slavery (however illusory it was, as in the case of 1860) was a threat to the Southern way of life. It would be overly simplistic, though not entirely unjustified, to portray Confederate armies as great mobs of impoverished, uneducated "white trash" (it's interesting that Grant states in his autobiography several times that this pretty much encompassed all working class whites) tricked into fighting for the interests of the wealthy. Still, the people who owned slaves were invariably the politicians who held power before the war and ultimately ratified the secession.
Generally, yes, the armies were made up of eligible voters (since, I'm pretty sure, most white men could vote since the Jacksonian era), though quite a substantial number were probably underage teenagers. Returning back to Grant's autobiography, he generally recognizes that Confederate armies had superb morale, trending toward a tendency for boldness and impetuosity in battles that, as time went on, could be worn down and cracked by superior Union discipline and determination.