Is DIE SPIDER DIE an interjection?

No, it is a complete sentence.

Die, spider, die is composed of two linked sentences: Die, spider and die. The subject of both sentences is an implied you (as if Die you).

An interjection is a part of speech that is interjected or popped into a sentence and which has no other grammatical meaning. For example, in Die, ugh, die, Spider, "ugh" is an interjection; and in I wanted the spider to die, but, oh, it just wouldn't, "oh" is an interjection; and in Ha! I killed the spider, "ha" is an interjection; and in Egads, the spider's still alive, "egads" is an interjection.