Can there be no complete subject in a sentence?

Answer 1

No, there has to be a subject in every sentence.

Not always, passive sentences don't have to have a subject.

Leonardo da vinci painted The Mona Lisa in the 16th century. - this is an active sentence we can see the subject is Leonardo Da Vinci.

The Mona Lisa was painted in the 16th century. - this is a passive sentence it doesn't tell us who (subject) painted the Mona Lisa.

The last comment is incorrect. In the example 'The Mona Lisa was painted in the 16th century', the subject of the sentence is 'The Mona Lisa'. It is the verb, not the sentence, that does not have a subject. That is the difference between active and passive constructions: in the former, the subject of the sentence is the subject of the verb; in the latter, the subject of the sentence is the object of the verb.

There are, however, constructions in which neither the sentence nor the verb has a subject. One is imperative sentences, for example 'Sit down!' Here the subject ('you') is understood but not stated. Another construction is impersonal verbs, which are typically used in statements about the weather, for example 'It is raining.' Here there is no subject; the whole sentence is a verb.

Answer 2

No, a complete sentence must have both a subject and a predicate. The subject is the part of the sentence that performs the action, while the predicate includes the verb and expresses what the subject is doing.