Guardo la cartera debajo de la almohada por si acaso.

Guardo is the present simple and can describe a habit, a general practice, or what you do in a situation, without focusing on “right now.”

• Guardo la cartera... = I keep / I put my wallet... (as a usual precaution, or as a general statement)
• Estoy guardando la cartera... = I’m putting my wallet away (right now) (emphasizes an action in progress)

Both, depending on context. Guardar covers:

• to keep/store: Guardo fotos en el móvil. (I keep/store photos on my phone.)
• to put away: Guardo la ropa en el armario. (I put the clothes away in the closet.)
  In your sentence it can naturally sound like either I keep or I put the wallet under the pillow (often implying as a precaution).

Spanish often uses the definite article (el/la/los/las) with personal belongings when it’s obvious whose it is.
So Guardo la cartera... commonly implies my wallet (or the relevant wallet in context).
You can say mi cartera for emphasis/contrast: Guardo mi cartera, no la tuya. (I keep my wallet, not yours.)

In Spain, la cartera commonly means wallet (where you keep money/cards).
For handbag/purse, Spain more often uses el bolso.
(Elsewhere, cartera can sometimes mean a bag/briefcase depending on region and context.)

Both exist, but:

• debajo de is very common and clear: “under/beneath”
• bajo can mean “under” too, but it can sound more formal or slightly different in tone, and it has other meanings (“low,” “short of,” etc.).
  So debajo de la almohada is the safest, most neutral choice.

Because debajo works like an adverb/noun meaning “underneath,” and it normally links to what it’s “underneath of” using de:

• debajo de + noun = underneath (of) something
  Example: debajo de la mesa (under the table)

por si acaso is a fixed expression meaning just in case / to be on the safe side.
It expresses a precaution without specifying a detailed condition.
Example: Lleva una chaqueta por si acaso. (Bring a jacket, just in case.)

Because the idea is a precaution in case something happens, not a straightforward purpose.

• para = purpose/goal: Lo hago para estar tranquilo. (I do it to be calm.)
• por si = in case (condition/possibility): Lo hago por si pasa algo. (I do it in case something happens.)
  por si acaso is the idiomatic, shortened way of saying that.

Yes. por si + clause is very common:

• Guardo la cartera debajo de la almohada por si viene alguien. (…in case someone comes.)
  por si acaso is often used when you don’t add a clause, or when you want a general “just in case” feeling:
• Guardo la cartera... por si acaso.

Sometimes, but differently. acaso on its own often appears in questions meaning something like by any chance:

• ¿Acaso crees que soy tonto? (Do you seriously think I’m stupid?/Do you by any chance think…?)
  In everyday “just in case,” it’s most natural in the fixed phrase por si acaso.

almohada ends in -a, and it’s feminine: la almohada.
There’s no special trick needed here; it behaves like most -a nouns.

Yes. Por si acaso can go at the beginning, middle, or end:

• Por si acaso, guardo la cartera debajo de la almohada.
• Guardo la cartera, por si acaso, debajo de la almohada. (more “parenthetical”)
• Guardo la cartera debajo de la almohada por si acaso. (very common)

No, because the verb ending -o in guardo already shows it’s I.
You add yo mainly for emphasis or contrast:

• Yo guardo la cartera debajo de la almohada, pero él no. (I do…, but he doesn’t.)