Learn Russian: Мэр назначил депутата председателем. - The mayor appointed the deputy as chairman.

This is the core rule for verbs of appointing, like назначить. The person being appointed takes the Accusative case (депутата), but the role, title, or position they are assigned to must take the Instrumental case (председателем).

Why does депутата end in "-а"?

It is in the Accusative case because it is the direct object (the person receiving the action). Since депутат is an animate masculine noun, its Accusative form matches its Genitive form, gaining the "-а" ending.

In English we say "appointed AS chairman". Do we ever need a preposition like как here?

No, you should not use как (as) with verbs of appointing. The Instrumental case on председателем already contains the meaning of "as" or "to the position of". Adding как would be a grammatical error in Russian.

What is the literal meaning behind the word председатель?

It comes from the prefix пред- (in front of / before) and the root -сед- (to sit). A председатель is literally someone who "sits in front" to lead a group, which perfectly aligns with the English concepts of a chairman or president.