Phrases such as Of course not can certainly stand alone, both in speaking and writing. But they are not fully-formed sentences of the type: simple, compound or complex.
The term for an incomplete sentence is sentence fragment or just fragment. The Cambridge Guide to English Usage (p317) has this extract in the entry on sentence:
In ordinary conversation, the sentences uttered may not be
fully-formed like those illustrated in section 1 above (i.e., simple,
compound or complex). When taking turns to speak, conversationalists often produce sentences whose wording builds partly
on the one before, without repeating every element.
The use of sentence fragments and elliptical connections between them
makes for highly efficient and highly cohesive discourse.
In your example, Of course not is an ellipsis of the fully-formed sentence:
Of course I am not to blame for the increase in deaths.
In answer to the edited question, a stand-alone subordinate clause (dependent clause) is also called a sentence fragment. The Longman Dictionary of Language Teaching and Applied Linguistics (p522) has this entry on sentence fragment:
(in composition) an incomplete sentence which cannot stand on its own.
Whenever I try to hold a conversation with my parents about my career is a sentence fragment because it is a dependent clause
which contains a subordinating conjunction and should therefore be
connected to an independent clause.