You seem to live a sheltered life. You'll be amazed by how many places there are where you can't use your phone because there's no signal. There's a National Park right in the middle of Rio de Janeiro where cell phones only work when the wind is blowing in the right direction ;-)
Or unless you live in California, the home of Silicon Valley and countless dead zones. Just last weekend, I was in the north Bay Area with zero cell signal. Even the middle of San Francisco is notorious for dropping signal. There are many instances when I cannot use my cell phone, but very few where I don't have permission to.
I think you are correct. Although poder can mean either, the correct translation in this context is "May I...", meaning "Am I allowed to..". Although "can I" (meaning able) is used widely, it is incorrect and first grade school teachers spend half the year trying to get the kids to say it right.
I used mobile, which was accepted (& has been many times in answers). In the UK this is a common term. I used celular when I have to write in Spanish. I am unsure why the term celular is being disputed. If we are learning Spanish & that is the word used there, then so be it. Try other words in translation that suit your country & see if they are accepted.
Cellular is an adjective (often abbreviated to "cell"), not a noun. It's not a thing -- it describes something. As in, cellular phone or cellular tower. Sometimes someone will say "Can I use my cell here?", but in that case, they are abbreviating "cell phone", not "cellular".
Yes, and I suppose "cellular" could technically abbreviate "cellular phone", but I've never heard anyone use it that way. With abbreviations being deviations from the rule, I think common usage is the most important factor. If someone said "Can I use my cellular here?", I'd know what they were saying, but I'd think it odd. In fact, even saying "cellular phone" is very rare in everyday speaking... it's almost always "cell phone". Although "cellular" by itself would be understandable to a fluent English speaker, I don't think it should be considered correct or accepted... after all, I can also usually understand a non-native speaker's broken English.
Usar is the infinitive form, which can translate in English as "to use", "using", or simply "use". After a modal verb like "can" you use the "bare infinitive" in English, meaning the base form without "to": "[someone] can use".
Uso is the form that's conjugated for yo, but you can't use that if you already have another conjugated verb in the same clause.
In the UK, at least, mobile phones have so far outstripped landlines/fixed phones that most people just call it a phone. Do a Google image search for “phone” and you see both kinds but the mobile variety are the commonest. In conversation I usually hear people saying “phone me” when they mean call me on my mobile phone (which might be their only phone) or “call me on my mobile” when they mean call them personally and not use the office number.
So, “phone” is now commonest, “mobile” is still used particularly to avoid confusion but “cell”, “cellular” and “cellphone” are for US movies.
Agree "phone", "mobile", "mobile phone", or, for the Americans, "cellphone" are appropriate translations, but surely not "cell"! Also will duolingo remind us that "celular" is not the only Spanish word for a mobile phone, and use "móvil" sometimes? Otherwise we won't be understood in Spain, but only in South America!