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Perhaps it gives a connotation of friendliness from "amigo". Wait did you see that, the French word for friend, just add "go" and it is Spanish!
Never mind apparently "amigo" is from the latin "amicus" and "conmigo" from latin "cum" and "mecum" while "contigo" is from latin "cum" and "tecum". http://lema.rae.es/drae/
“bailar" is the infinitive form of the verb, which is what is required. Think of it this way: in English, the infinitive form would be “dance" for the action “to dance". You don't ask “do you want to danced/to dancing..."etc. You just go with the infinitive form. This is easier in English, but i hope you get the point.
Same in Spanish. You need the infinitive form. Not “quieres bailamos" or “quieres bailavan" or any other conjugation. The word “to" before the verb is a good indication that the infinitive form will be correct.
It is not in your conjugation table specifically because by definition the infinitive form is the base verb in its unconjugated form.
De nada. In Spanish, the infinitive forms of verbs will end with -ar, -ir, or -er. I also should make a minor correction. The English infinitive form would be "to dance", not simply "dance". In English, the infinitive form always includes "to" + verb. In Spanish, it is just the one word verb which ends with -ar, -ir, or -er.