Translation:Do you have a cell phone?
Speaking of スマホ, the following commercial immediately comes to mind: https://www.youtube.com/watch?v=0bUnqXwrQ30 . What the samurai says is "映画はスマホで", which means "(You can watch) movies on the smartphone". Besides that one, the SoftBank commercial series is also worth watching (my favorite commercial from it is https://www.youtube.com/watch?v=45tK7szJFx0 )
Yes. It can mean exist, but it can also mean do you have and is very commonly used in this sense. I would use aru in general to ask someone if they have/own something and motte iru to make sure, for instance, a friend hadn't forgotten to bring something with them eg. Pan arimasu ka is there any bread OR do you have any bread? Kippu motteru? Have you got the tickets?
You must be forgetting that this is a high-context language. When my teacher would ask "質問(しつもん)がありますか?", she wasn't asking if questions were a thing that existed somewhere; she was asking if we had any questions.
Likewise, no one would interpret "携帯ありますか?" as "do cellphones exist?", because that's absurd.
It may be because "have you got..." is technically improper grammar in English, and Duo sometimes gets picky about that IME. I kept getting a different task wrong because I was using "loan" as the verb instead of "lend", for example.
But I still think you should report it, because I ain't no language prescriptivist :)
Yes, it seem that many people try to make that point here. But that doesn't change the fact that 携帯電話 is still the correct word. In context or unofficial situations you'd use 携帯 or 電話 just because it's shorter. But 電話 is every phone, 携帯 can be almost any mobile device (I've heard people refer to their Nintendo DS with 携帯) and スマホ is Smartphone (if you want to borrow a phone you don't explicitly ask for a Smartphone). So it's good that people point out that the other versions are more often used in modern Japanese, but (for clarity) 携帯電話 is still the correct, arguably unwieldy, but (in my eyes or experiences) not dated word.
It would sound a little odd - like you were asking someone if they were literally holding a cell phone in the hand, right now. Do you have your cell phone or do you have a cell phone (ie. on you but not necessarily in your hand) would sound more natural and convey the right meaning.
A mobile phone could mean a portable cordless (landline) phone or it could also mean a cell phone. Depends where you live - country, area etc. The same words can have different meanings - sometimes vastly different - depending on what country you live in or even what part of the country you live in. Where I'm from in NZ we say I'm vacuuming or hoovering the house when cleaning the house with a vacuum cleaner, but further south (same island) they say I'm luxing - because Electrolux is a vacuum cleaner brand name.