هاتفي جديد، وأكتب رسالة إلى أمي الآن.

The ending ـي is a attached possessive pronoun meaning my.

• هاتف = phone
• هاتفي = my phone (هاتف + ـي)
  In writing, the word is connected and you don’t add a separate word for my.

Common pronunciation (MSA, without full case endings) is roughly: hā-ti-fī.

• ها = hā (long ā)
• تِ = ti
• في = fī (long ī)
  The final ـي is a long ee sound.

Arabic often uses a nominal sentence (a sentence with no overt verb) in the present tense.
So هاتفي جديد literally is My phone (is) new with is understood.

Because here جديد is the predicate of the sentence (My phone is new). In Arabic, predicates in such “X is Y” sentences are very commonly indefinite (no ال), even if the subject is definite.
If you said هاتفي الجديد, that would normally mean my new phone (a noun phrase), not a full “is” sentence.

Yes. هاتف is grammatically masculine, so the adjective is masculine singular:

• masculine: جديد
• feminine would be: جديدة (used with a feminine noun)

و means and, and it links two clauses:
1) هاتفي جديد
2) أكتب رسالة إلى أمي الآن
It can connect sentences even if the subjects are different or only implied.

، is the Arabic comma. It’s used similarly to English, though punctuation can be more flexible in Arabic texts. Here it separates two related parts before adding another clause with و.

Because Arabic verb forms include the subject. أكتب is the 1st person singular imperfect form, so I is built in.
You can add أنا for emphasis or contrast, but it’s not required.

It’s the imperfect tense, which can cover:

• habitual: I write
• current action: I am writing
  The word الآن (now) makes it clearly right now here.

رسالة is the direct object of أكتب (I write a message).
In fully vowelled MSA, it would often be رسالةً (accusative + tanwīn), but those endings are usually omitted in normal writing.

إلى means to, and it introduces the recipient: to my mother.
Also, after a preposition like إلى, the noun is grammatically in the genitive (majrūr). With full vowel marks you might see something like إلى أُمّي.