Could you also say "Bayramınız kutlu olsun" by analogy with "yeni yılınız kutlu olsun"?

Yes, "Bayramınız kutlu olsun" is more common. At least that's what we say and what we say to friends and family who come around.

You usually kiss the person's hand and put against your forehead if the person you are saying it to is your elder (not older by a few years, OLDER). It's respectful. Otherwise kissing on each cheek is fine.

Thanks for explaining Teanga