Learn Croatian: Upotreba mobitela nije dozvoljena. - The use of mobile phones is not allowed.

Because smjeti is a verb that pairs with another action (like 'you are not allowed to use'). In this sentence, the subject is the noun upotreba (the use). Since we are describing a noun, we use the adjective dozvoljena (allowed) alongside nije (is not).

How could I rephrase this sentence to use ne smjeti, which we are learning in this lesson?

You can use the impersonal se construction. You would say Ne smije se upotrebljavati mobitel (One must not use a mobile phone) or Mobiteli se ne smiju upotrebljavati (Mobile phones must not be used).

Why is the word for mobile phone written as mobitela?

It is in the genitive case. Whenever you have a 'noun of a noun' structure, like 'the use of a mobile phone', the second noun takes the genitive. For the masculine noun mobitel, the genitive adds an -a. It translates equally well to 'of mobile phones', as the genitive plural form is also mobitela.

Why does the word dozvoljena end in -a?

Dozvoljena functions as an adjective here and must agree with the gender of the subject it describes. The subject is upotreba (use), which is a feminine noun, so the adjective takes the feminine ending -a.