"She has a cute cell phone."
I was also curious and found the following quote online:
It all boils down to the context of the situation. Most of the time, when you having a conversation the context of the conversation is VERY EASILY understood and the meaning of a sentence is obvious.
If someone out of the blue asks you "Terebi ga arimasu ka" or "Terebi o motteimasu ka" they both mean "Do you have a TV". If you were holding something you wouldn't be asked "Terebi ga arimasu ka" but might be asked "Terebi o motteimasu ka", to mean "Are you holding a TV?".
Which hopefully answers your question! So you should be able to say either for this question without proper context.
you would have to say もっています for it to mean "she has," because かのじょ is the subject of もっています. あります is an intransitive verb, so けいたい電話 would become the subject of it. けいたい電話があります would be more like "there is a cell phone." Wouldn't really make sense if you wanted to say "She has a phone."
It's not a stupid question! Duo doesn't explain this very well and kind of relies on you knowing about the て form...
You're on the right track already with your question as you're right, it does end in います！the te before hand is part of the conjugation of the verb 持つ "to have/hold/own". In theて form (持って) it becomes something you currently have/hold/own.
There's more info here which might be explained better: http://yesjapan.com/YJ6/question/4142/what-is-the-difference-between-motte-imasu-and-arimasu-imasu