Mi gato duerme debajo de la cama.

Mi is the possessive adjective that goes before a noun: mi gato = my cat.
Mío is a possessive pronoun (or stressed possessive) and is used differently, usually without a noun or after the noun for emphasis:

• El gato es mío. = The cat is mine.
• Mi gato (neutral) vs. el gato mío (less common; can sound emphatic or regional).

Because the subject is mi gato (he/she/it), so you use the third-person singular form of dormir in the present tense:

• yo duermo = I sleep
• tú duermes = you sleep
• él/ella/usted duerme = he/she/you (formal) sleeps
  Here, gato is treated as él/ella depending on the cat’s sex.

Dormir is a stem-changing verb (o → ue) in most present-tense forms. The o in the stem changes when stressed:

• yo duermo, tú duermes, él/ella duerme, ellos duermen
  It does not change in nosotros/vosotros:
• nosotros dormimos, vosotros dormís

Grammatically, gato is masculine. If you want to specify a female cat, you typically say gata:

• Mi gata duerme debajo de la cama.
  In everyday speech, some people still say mi gato even for a female cat (especially if they’re used to the pet’s name), but gata is the clear option.

In Spanish, debajo is commonly followed by de + noun to show under something: debajo de la cama = under the bed.
Debajo alone usually needs context (like down below):

• Está debajo. = It’s underneath/down below. (but not as specific as debajo de...)

Spanish often uses the definite article (el/la) with common objects in a known or typical setting, especially in “location” phrases: debajo de la cama sounds like the bed (in that room / the usual bed).
You’d use una cama if it’s not specific or is being introduced:

• Mi gato duerme debajo de una cama que encontré en el sótano. = ...under a bed I found...

Usually no. Spanish often drops subject pronouns because the verb ending shows who the subject is.

• Duerme debajo de la cama. can mean He/She/It sleeps under the bed if the subject is clear from context.
  Here, Mi gato duerme... already states the subject, so no pronoun is needed.

It’s most natural at the end, but Spanish allows flexibility:

• Mi gato duerme debajo de la cama. (most neutral)
• Debajo de la cama, mi gato duerme. (more emphasis on where)
  The second sounds more “storytelling” or contrastive.

No—bajo de is generally not correct for “under.” You want either:

• debajo de + noun (very common)
• bajo + noun (also correct)
  So: debajo de la cama or bajo la cama.

Often Spanish uses the present tense for both “sleeps” and “is sleeping,” depending on context:

• Mi gato duerme debajo de la cama. can mean sleeps or is sleeping.
  If you want to emphasize “right now,” you can use estar + gerundio:
• Mi gato está durmiendo debajo de la cama.