Fais attention à ne pas oublier ton parapluie.

Because the negative ne…pas in this sentence applies to the infinitive oublier, not to the imperative fais attention.

• Fais attention means “pay attention” or “be careful.”
• à ne pas oublier literally “to not forget.”
  If you put the negation around fais attention, i.e. Ne fais pas attention, it means “don’t pay attention,” which is the opposite of what you want.

The verb phrase faire attention governs the preposition à when it’s followed by an infinitive or noun.

• Correct: faire attention à
  • verb/noun
• Old-fashioned or literary: faire attention de exists but is rare in modern French.
  Therefore you always link faire attention with à, even when you insert the negative.

Yes.

• N’oublie pas ton parapluie = “Don’t forget your umbrella.”
  That’s shorter and perfectly idiomatic.
  Fais attention à ne pas… adds a nuance of warning or caution (“be sure not to…,” “take care not to…”).

Because the imperative fais is singular and informal (tu-form).

• With “tu,” you use ton (masculine singular).
  If you addressed someone formally or a group, you’d say:
• Faites attention à ne pas oublier votre parapluie.

Because parapluie is a masculine noun (un parapluie).

• Masculine singular → ton.
  If the noun were feminine (e.g. ma valise), you’d use ta.

Yes:

• faire attention à ne pas… = “take care not to…,” emphasizes caution.
• penser à = “remember to…,” emphasizes recollection or memory.
  Example:
  • Fais attention à ne pas glisser. (“Be careful not to slip.”)
  • Pense à prendre tes clés. (“Remember to take your keys.”)

Informally, yes.

• Fais gaffe = colloquial for “watch out” or “be careful.”
  But note register:
• Fais attention is neutral.
• Fais gaffe is very casual, spoken style.