'Do you want to' is a neutral question that shows no clear bias towards any possible reply. You're genuinely asking someone what they themselves actually want.
And 'please' (the literal translation of 'være så snill' is 'be so kind') is a marker of an offer - meaning, you display clear bias towards the positive reply, and if the person you're asking refuse to dance with you, they know you'll be disappointed.
This means that you generally don't want to use both at the same time. Putting 'please' into a 'do you want to' question makes it sound disingenuous. It's like you're threatening someone under the guise of an innocent enquiry.
So, with this conundrum in mind, since there is no way to misinterpret the 'være så snill' part of the original sentence, the expected conclusion is that we're supposed to resolve the ambiguity by translating 'vil du' as 'will you' and not as 'do you want to'.
You can think of the phrase "Vil du være så snill å.." as the same as the english phrase "would you be so kind as to.."
In normal Norwegian conversation there is no substitute for a simple "please" as is used often in English, but you could insert this phrase if you really want to try and convince someone to do something for you, like washing the dishes, or in this case asking for a dance.
You'd only ever use this if you were trying to be formal/courteous or if you were a parent talking to a child.
"(å) være (så snill)" is the infinitive. It's used because of the "vil du" in front of it. After verbs like "kan" and "vil" (I think they are modal verbs, correct me if I'm wrong) you always have to use the infinitive form. "vær så snill", on the other hand, is the imperative form. The sentence "Vær så snill og dans med meg." has two imperatives and is a somewhat more direct approach to the same request.