Is there a difference between a very polite and informal way to say you want an ice cream? The sentence the way it is, it seems very formal and polite to me.
The sentence above is polite, but I wouldn't say it's very formal and polite. Since we don't use "please" that much, this "skulle vilja ha" is quite common.
What are the most polite and most informal versions of the same sentence?
The most polite is maybe "Skulle jag kunna få en glass, tack?" (Could I please have an ice cream?) and the most informal is "Hit med en glass!" but that is very rude :).
Forgive me for resurrecting a long-quiet question, but I've been trying to get the scale of politeness in my head as well. How does this look? (I've put what I think are very literal English translations, to help me remember them)
Formal: "Skulle jag kunna få en glass, tack?" (Could I be able to get an ice cream, thanks/please?)
Polite: "Jag skulle vilja en glass" (I would choose an ice-cream)
Casual: "Jag vill ha en glass" (I want [to have] an ice-cream)
Demanding: "Ge mig en glass" (Give me an ice-cream"
Appropriate if you are mugging someone: "Hit med en glass" (Hand over the icecream)
Your polite form is missing the ha in vilja ha, and I wouldn't call the formal version formal really - just a little bit more polite. Otherwise, your Swedish is spot on.
In this case, how would you know whether she is asking for an ice cream or a glass? When it's written, sure, but in an out-of-context sentence like this, I really have to strain to hear whether she is using a long or a short vowel.