The word scaredy doesn't exist on its own. The phrase scaredy cat
first appears in print in Dorothy Parker's The Waltz, published in
1933, and she's usually credited with coining the term.
Thus, it is used with the word cat attached to it.
Though, there has also been another cartoon titled Scaredy Squirrel, but it still isn't used for the phrase.
Yes, it is used explicitly to depict someone as timid or refer to someone who is afraid/coward. I'm not sure that it is applicable to one's shyness too.
Your usage of
scaredy-cat in the following
He is a scaredy cat.
is perfectly correct. Yes, it isn't polite to call someone a coward; but it is correct.