Would Vær så snill og bruk telefonen also work? Which is more common?
Yes it would! They're both fairly common, although you tend to see "vennligst" more on signs and written materials.
I have never seen this sign in Norway, where is it and what does it mean?
In conversation, which is more common and preferred?
What's the difference of usage between vær så snill and vennligst?
'Vennligst' is used in a more formal way, on signs, in writing, essentially means "Kindly" from what other native speakers have said. 'Vær så snill" means "please". Translating word-for-word it means 'be so kind' but colloquially, it means 'please'.