What you have here isn't a list of function but a list of list of symbols. Containing also some symbols... Since you quoted everything, then everything is quoted and only numbers and lists, strings are self-quoted. If you want to keep objects inside a list, you have two choices here:
Create a list instead of quoting it.
(define f
(list (lambda (n) (+ n 2))
(lambda (n) (* n 2))
(lambda (n) (* n n))))
Pseudo-quote it and unquote each function declaration
(define f
`(,(lambda (n) (+ n 2))
,(lambda (n) (* n 2))
,(lambda (n) (* n n))))
To be honest, the explicit list seems a bit nicer than the pseudo-quote version.
What your quoted version gives you is this:
(define f
(list (list 'lambda (list 'n) (list '+ 'n 2))
(list 'lambda (list 'n) (list '* 'n 2))
(list 'lambda (list 'n) (list '* 'n 'n))))
Quote isn't a shorthand for makings lists. It's a shorthand to not evaluate anything.