Learn English – “everybody” vs “everybody else”

In modern English, genitive 's is a clitic that attaches to an entire noun phrase, as in example 1 below.
It doesn't attach to an individual noun, as in example 2:

 1. [ The King of England ] 's pants
 2. *The King's of England pants　　　　(ungrammatical)

The postmodifier else can attach to an indefinite pronoun like somebody, as in example 3 below.
It can't attach to a genitive noun phrase like somebody's, as in example 4:

 3. [ somebody else ]
 4. *[ somebody ] 's else　　　　(ungrammatical)

Since somebody else is a complete noun phrase, 's can attach to it:

 5. [ somebody else ] 's

This is the correct choice.

It's not a pronoun because it doesn't take the place of a noun - not by itself anyway. No one/nobody/nothing/nowhere would be some of the "pronoun forms" of no.

No generally modifies a noun so that makes it technically an adjective. When modifying a verb or adjective not is used.

No is also a typically a determiner - which can be considered a special type of adjective - because it can occupy the "spot" where an article or word like this/that/these/those would go.