Usually we don't use the subject because, as you said, "quiero" already tells you the subject of the sentence. You would use the subject only when you want to emphasize it for some reason. For instance: You are at work and the boss tells you and your coworkers to come to work on Sunday, so you stand up and says "YO no quiero trabajar el domingo". You emphasize that fact that YOU don't want to come to work. In any other situation seems odd to use the subject.
In English, "I don't want work" doesn't have exactly the same meaning as "I don't want to work." For example the first sentence could mean something like you don't want to receive more homework.
However it seems the Spanish sentence could mean either and should be accepted. They seem to want you to use the infinitive "to work" "trabajar".
In Spanish, when two or more verbs appear consecutively one after the other, the first is always conjugated for the subject while all the rest are left in their infinitive forms. The rule is exactly the same as in English; after all, you don't say "I don't want I work", right? Instead you say "I don't want to work".