"coming" is used instead of "come" because the "present continuous tense" (is coming) is used in this sentence to replace the "simple future tense" (will come), to describe an action in the future.
Therefore, "She is not coming home tonight." = "She will not come home tonight".
The usage of "present continuous tense" instead of "simple future tense" for a future action is becoming more popular, especially in informal or conversational English.
However, "She does not come home tonight." does not describe a future action, so "come" is not used.
If we want to describe a fact or usual event, we will use "come". For example: "She does not come home on Monday nights".
"She does not come home any more."
She is not coming home tonight.
coming is a participle.
participle n,(grammar) in English is a word formed from a verb, ending in ---ing(= the present participle of come.)
Hence:-Present Infinitive--come
Past Tense---came
Past Pasticiple---come.
Irregular verb=verbs conjugated irregularly.
One cannot find here when
--ed, --en, etc. (=the past participle)
where participial is the adjective.