Yes, "he's afraid of his own shadow" is a common expression.
For the first sentence, you should either say
He's very cowardly
OR He's a real coward
OR He's so cowardly that he's afraid of his own shadow
OR He's such a coward that he's afraid of his own shadow
"coward" is a noun
"cowardly" is an adjective (yes, -ly USUALLY turns adjectives into adverbs, but there are a few words where it turns nouns into adjectives, like "cowardly" and "slovenly")
He is so [adj]: He is so ugly that he scares dogs. He is so mad that he can't think.
He is such a [noun]: He is such a hero that he'll risk his life to save a mouse. He is such a genius that he's smarter than Einstein.
He is such an [adj] [noun]. He is such a fat man that he broke my chair.
If you use "so" or "such" in writing, put "that" after it.
He was so sad that he cried for an hour.
It was such a salty meal that he couldn't eat it.
Using "so" or "such" without "that" is very, very casual. In writing, you should use "very".
spoken: "He's so sad!"
written: he's very sad.