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I don't think that the last point is relevant, no. There is normally a range of screen timeout or sleep settings....although I don't know why 'Never' is not an option for you. But I don't think that that is where the problem lies.
The first thing to try, if you are able to boot into Recovery, is a Cache Partition wipe. That causes no loss of personal data, and it may help.
Let us know how you get on.
Regarding terminology, as I understand it a mobile phone is a basic voice-and-messaging phone that works by connecting to a radio network, which is built up of small areas or cells, a bit like a honeycomb. A smartphone is the same, but uses higher frequencies, allowing it to connect to the internet as well. So both mobile phones and smartphones are cellphones.
Here in the UK the term 'cellphone' is not really used.
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