Why is 面熟 used after 人 in 常常看着一个人面熟却叫不出名字来?
This is a classic example of a topic-comment construction that is prevalent in Chinese.  In this case, 面熟 is not serving as an adjective to the noun, but rather as a comment on the topic.


常常看着一个人 ("often seeing a person") is the topic
面熟却叫不出名字来 ("[he's] familiar, yet [I] can't come up with [his] name") is the comment.


All adjectives in Chinese can function as verbs (Chinese adjectives are sometimes called stative verbs for this reason); as such, they can function as a predicate by themselves.  Because Chinese is a pro-drop language, predicates can serve as their own clauses in a comment.

To answer your second question, you can't put 面熟 before the noun in this case without restructuring the sentence.  却 is used to connect two clauses together.  If you make 面熟 into an adjective, it's no longer a clause, so the usage of 却 will no longer make sense.