[noun] + など / なんか / なんて [noun] + や + [noun] + など [verb (plain)] + など [sentence] + なんて
など, なんか and なんて go after a noun and mean "things like", "the likes of". など is the neutral, written one, and it lists examples. なんか and なんて are casual, and they usually belittle the noun they follow – yourself included.
など closes a list of examples, usually joined with や: "A, B and so on". The list is not complete.
In casual speech なんか does the same job, and it softens a suggestion: こうひいコーヒーなんかどうきごう?, "how about a coffee or something?"
After a noun, なんか and なんて push it down: "something like that", "the likes of". With 私 it is modesty; with anything else it is dismissal, and a negative usually follows. In polite or written Japanese, など does this.
Only なんて can follow a whole clause. It marks what you just said as surprising or hard to believe, and your reaction comes after it.
These go beyond "things like".
At the start of a sentence, なんか is a filler, like English "kind of": the speaker is being vague on purpose. This なんか comes from なんか何か, "something".
In なんていった言ったきごう?, "what did you say?", なんて is a casual 何と: "what", not "things like". The exclamation なんて, "what a…!", is a separate point.