Su mirada expresa alegría, y la recuerdo todo el día.

In Spanish, su is a possessive adjective that can mean “his,” “her,” “your” (formal), or “their.” Here you need context to decide:

• If you’re talking about a woman, su = “her.”
• If it’s about a man, su = “his.”
• If it’s formal speech to someone, su = “your.”
• If you’re speaking about a group, su = “their.”
  Since the sentence doesn’t specify, learners often interpret su as “her” or “his,” depending on surrounding text.

Spanish often omits the definite article when talking about abstract nouns or general concepts. Here alegría means “joy” in an abstract sense:

• Su mirada expresa alegría = “Her gaze expresses joy,” not a specific joy.
  If you wanted a specific, previously mentioned joy you could say “la alegría,” but that isn’t needed for a general feeling.

The subject of the verb is “su mirada” (her gaze), which is third person singular. In Spanish you conjugate expresar accordingly:

• yo expreso
• tú expresas
• él/ella/usted expresa ← matches su mirada
• nosotros expresamos, etc.

La is a direct‐object pronoun replacing a feminine noun. Here it stands for la mirada (feminine singular):

• Recuerdo la mirada → La recuerdo
  So instead of repeating la mirada, you use la.

Yes. Durante todo el día also means “throughout the day.”
Both are correct; durante simply makes it explicit:

• la recuerdo todo el día
• la recuerdo durante todo el día

In Spanish you can link two clauses with y (and) without que. Adding que would change the meaning:

• …y la recuerdo todo el día (I remember it all day)
• …y que la recuerdo todo el día would sound like a subordinate clause, shifting emphasis or introducing reported speech. It’s more natural here to keep it simple.

Both can mean “I remember,” but:

• recuerdo is transitive, so it takes a direct object: recuerdo la mirada → la recuerdo.
• me acuerdo (de) is intransitive, requiring de: me acuerdo de la mirada.
  You cannot say me recuerdo la mirada (incorrect). Use recuerdo
  • object or me acuerdo de
    • noun.