Something can be both a phrase and a sentence fragment, but they aren't the same thing.
Gramatically a phrase is a group of words that functions as a unit in the syntax of a sentence. So in the sentence "A vanilla ice cream with sprinkles looks tasty," "with sprinkles" is a prepositional phrase, "A vanilla ice cream" is a noun phrase, and "A vanilla ice cream with sprinkles" is a larger noun phrase. I can use any of these phrasal units to describe parts of the sentence. We could also talk about verb phrases - "looks tasty" is a verb phrase formed from the main verb and its complement.
A sentence fragment is any set of words that don't form a complete sentence. Maybe it's missing a syntactic element like a subject or verb; maybe it doesn't express a complete thought. In the example above, all of those phrases could be considered incomplete sentences if they appeared alone, but I could also form sentence fragments that are not phrases, like "looks" or "tasty." That said, when someone labels something a sentence fragment, it's usually because they're expecting a complete sentence.
To take your examples:
hit a nerve - could be read as an imperative sentence ordering someone to hit a nerve, so it's not a sentence fragment unless the context makes it clear that a subject is missing. It can be read as a verb phrase formed from the verb "hit" and the noun phrase "a nerve."
keep it safe - same as before. You're just adding a bit more complexity.
let yourself in - same as before.
find the keys to the door - same as before
useless detective can't catch the criminals. -can be read as a sentence with a subject ("useless detective"), verb ("can't catch"), and direct object ("the criminals"). We have multiple phrases forming this sentence. In this reading, unless context suggested otherwise, I'd say that there's a missing article: "The useless detective" or "A useless detective" "can't catch the criminals."
find out who else can take care of the dog. - see the first response. This one only differs in how many smaller components form the verb phrase: "the dog" (NP) -> "of the dog" (PP) -> "take care of the dog" (VP) -> "who else can take care of the dog" (NP) -> "find out who else can take care of the dog" (VP).
assume his identity is real - same as before.