Non dimenticarti di scrivermi appena arrivi.

Dimenticare is a transitive verb (“to forget something”): you can say Ho dimenticato le chiavi (“I forgot the keys”).
Dimenticarsi is a pronominal (reflexive) or intransitive form used when you “forget to do something.” It always takes di + infinitive afterward. In the 2nd person singular informal negative command, Italian uses non + infinitive, so:

• Non dimenticare (di…)
• Non dimenticarti (di…)
  The -ti is the reflexive pronoun “yourself,” attached to the infinitive dimenticare because it’s a negative imperative.

Verbs like dimenticarsi, accorgersi, ricordarsi require di when introducing another verb in the infinitive. The pattern is:

• dimenticarsi di
  • infinitive
    So you must say Non dimenticarti di scrivermi, not dimenticarti scrivermi, otherwise it’s ungrammatical.

Because the governing verb here is in the infinitive (“di scrivere”), clitic pronouns (mi, ti, lo, ecc.) attach to the end of the infinitive: scrivere → scrivermi. In Italian:

• after an infinitive or with non-imperative forms you attach clitics (scrivermi, leggerlo, ecc.).
• with finite verbs (io ti chiamo, lui mi aspetta) the clitic precedes.

Appena is a subordinating conjunction meaning “as soon as.” You don’t add che after it in modern usage (though older/written styles sometimes had appena che). You could replace it with non appena without changing the meaning:

• Non dimenticarti di scrivermi non appena arrivi.

Yes. Italian allows you to front the temporal clause:

• Appena arrivi, non dimenticarti di scrivermi.
  Both orders are natural; fronting appena arrivi can add emphasis to the timing.