"I need a cellphone."
Translation:Ich brauche ein Handy.
In spoken conversation it would be fine, for written comments better use standard German. Perhaps it might be accepted as "ich brauch'". Duolingo calls it a mistake if you have a word that is written like the one you used. In this case, it is "der Brauch", the custom/the tradition.
Yes, but to date the two-word version is preferred. Duo's acceptance of both versions from users is a good thing, but ideally Duo itself would use only the preferred spelling. In other words, Duo should not distract learners by slapping their hands when they use a recognized alternate spelling in English, since they're not making a mistake in German, but Duo should still set an example by sticking to Standard English itself, the more so since not everyone doing German-for-English-speakers actually speaks English as their first language.
Believe it or not, people still use landline phones, especially in business contexts. A (tele)phone can be a cellphone or a landline phone. The word (tele)phone is to the word cellphone what the word pet is to the word dog, i.e. generic vs. specific.
the tele(phone) = das Telefon
the cellphone (the mobile phone) = das Handy / das Mobiltelefon
Regardless of whether you always refer to your cellphone as a phone, in this course, you will not be able to use (tele)phone/Telefon and cellphone/Handy interchangeably.