Learn Arabic (Modern Standard): الرسالة رقم خمسة من المدير.

Because we are not counting a quantity of messages here. Instead, we are using the number as an abstract label or ID. When reading numbers as plain digits in this bare counting form, Arabic always uses the default version ending in a taa marbuuta, like خمسة, regardless of the noun's gender.

Why does the sentence say الرسالة رقم (the message number) instead of just message number?

In English, saying message number five makes it specific automatically. In Arabic, because you are talking about a specific message, you must attach the definite article ال to the noun. It translates literally as: the message, number five.

Where is the Arabic word for is in this sentence?

Arabic does not use a word for is or are in the present tense. You simply state the subject (الرسالة رقم خمسة) followed directly by the information about it (من المدير). This forms what is known as a nominal sentence.

Does the word for from (من) affect the word for manager (المدير)?

Yes, من is a preposition. In formal Arabic grammar, prepositions make the noun after them end in a short i sound, making it المديري (al-mudiiri). However, it is perfectly fine for beginners to drop this ending and just pronounce it المدير (al-mudiir) as most everyday speakers do.