Thank you for the questions, Emma.

>1. Is the word for "and" and
>"also" one and the same: ok?

Yes.

>and is it pronounced like the tree - oak,
>or the bird - ok?

Something in between:) like French 'oc'.

>2. Syntax: it appears that it is accusative, verb, nominative. Yes?
>What if you have Sven is a King? Both Svenr and Konungr are
>nominative, yes? If so what is the syntax? Svenr er konungr. Or would
>you instead say: heitir konungrinn Svenr.

The word order is fairly free; you can have
subject-verb-object, as in English, or something
quite different.

Heitir konungrinn Sveinn.
Sveinn heitir konungrinn.
Konungrinn heitir Sveinn.

>3. Complements: is there an equivalent complement between
>accusatives? For example, if you said: The King kills the elf, also a
>king. Would that be: Alf vegr konungrinn, ok konunginn.

I'm not quite sure what you're getting at,
maybe you meant 'subject' rather than complement
but the sentence:

"Álf vegr konungrinn ok dverg."

Is allowed and means:

"The king kills an elf and a dwarf."

Regards,
Haukur