A democracy means everyone is equal before the states, not anything goes. Meaning if you have one right, so do the other. That's simple but it's to avoid conflict/favoritism that would end up in conflict. The reason polygamy isn't allowed is noone is allowed to have several wives/husbands(so everyone is the same in regards to that). Polygamy/polyandry isn't allowed because noone pushes for it and I guess it would be an hassle to administratively. It could be allowed if someone proposed a law with the proper framework but noone bothered. As for minors, they can't enter contracts that requires informed consent, it's not specific to marriage.
I brought up the constitution to highlight what a democracy is and to show that it isn't just about voting this or that. As for the reason marriage for homosexuals should be allowed:
-allows the people concerned the same chances to happiness
-insurance
-adoption
-inheritance
-etc... (everything heterosexual couples are allowed to have under a marriage contract)
-equal rights
Downsides:
-some people don't like homosexuals
I guess what I said was a bit off topic since the OP is "Hate the sin but love the person...except when voting?" but it seems you've shown the only reason you don't want homosexuals to marry is because you don't like it. You're just confirming the suspicion of the OP that you dislike the "sinner" so we have yet to meet one person who "hates the sin but loves the sinner".