My character wants to marry Serana. Will he have to become a vampire, or will she have to take the cure? Can a mortal marry a vampire?

Also, can I marry a character who can't be married?

You can only marry Serana on PC with a mod.

You can't marry Serana without mods.

If an NPC can't be married, but shares a voice actor with an NPC that can then you can use console commands to make them marriageable.

http://www.gamefaqs.com/boards/615805-the-elder-scrolls-v-skyrim/61242343

Unfortunately, Serana can't be married. If you have a PC, fortunately, there are a lot of mods that make her marriable. A LOT.

Than you for the above replies, they are very helpful. But lets say I have made her marriageable, could a mortal marry her and have her stay a vampire or would they both have to be alive/undead?

You don't have to be undead or have her alive if you marry Serana.

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That, my children, is a first sign of Necrophilia.