three sentences

1) You dance to music, you don't dance on music.

2) It means to set something up, usually electronic things. Can you rig up a P.A. system? He rigged up a computer network. Etc.

3) "as he watched them leave". Keep the past tense throughout this sentence, because the presence of "as" doesn't change anything or require a gerund, if that's what you were thinking.

//**edit: Isn't it "Rocking Around the Christmas Tree"? I think so, but maybe one of the gentiles should confirm this.

Confirmed by a gentile.

3) He smiled triumphantly as he watched them leave.

or...He smiled triumphantly as he watching them leaving.

or...He smiled triumphantly as he watching them leave.

3)

Though it might not sound as good, because of the tense thing, you could play around with adverbial phrases and phrase reductions a bit in this sentence.

1. He smiled triumphantly, while watching them leave.

2. Watching them leave, he smiled triumphantly.

3. As he watched them leave, he smiled triumphantly.

4. Smiling triumphantly, he watched them leave.

Each sentence changes the focus of the sentence a bit, but they are all perfectly acceptable.