Depois da corrida, ela bebe um iogurte frio.

Because in Portuguese the preposition de contracts with the definite article:

• de + a = da
• de + o = do
• de + as = das
• de + os = dos

Since corrida is feminine singular (takes a), you get depois da corrida. This contraction is mandatory in standard Portuguese.

Yes, both are correct but they nuance the idea differently:

• Depois da corrida = after the run/race (a specific event)
• Depois de correr = after running (the activity in general)

For a general habit, Depois de correr, ela bebe… sounds very natural. If you want a non-specific event with the noun, you can also say Depois de uma corrida…

• um iogurte uses the indefinite article because the yogurt is not specific (it’s “a yogurt,” any one).
• da corrida uses the definite article because it refers to a specific run (the one just finished).

To make the run non-specific you could say depois de uma corrida (“after a run”).

In European Portuguese, the simple present often expresses a habit or routine: Ela bebe… = “She drinks (as a habit).”

• One completed past occasion: Ela bebeu…
• Ongoing right now: Ela está a beber…
• Near future: Ela vai beber…

It depends on the type:

• Spoonable yogurt: more often comer um iogurte (“eat a yogurt”).
• Drinkable yogurt: beber um iogurte (“drink a yogurt”).
• tomar (“have”) is also common in EP for food/drink/medicine: tomar um iogurte, tomar um café, tomar comprimidos.

Not exactly.

• frio = cold/chilled.
• gelado = very cold/icy; and as a noun in Portugal, gelado means “ice cream.”

So iogurte gelado could be understood as “frozen yogurt” (the dessert) or “very cold yogurt,” depending on context. For a normally chilled yogurt, iogurte frio (or bem fresco/bem frio) is safer.

Yes. Word order is flexible:

• Depois da corrida, ela bebe um iogurte frio.
• Ela bebe um iogurte frio depois da corrida. Fronting the time phrase adds emphasis to the time and flows very naturally.

Approximate IPA and tips:

• depois [dɨˈpɔjʃ]; in connected speech with da, you’ll often hear the linking: depois da [dɨˈpɔjʒ dɐ] (the final s in depois becomes a voiced “zh” before the d).
• da [dɐ]
• corrida [kuˈʁiðɐ] (the double r is a throaty [ʁ] in most EP accents)
• ela [ˈɛlɐ]
• bebe [ˈbɛbɨ] (final e is a very short, “uh‑like” sound)
• um [ũ] (nasal)
• iogurte [juˈɡuɾt(ɨ)] (initial io sounds like English “yoo”)
• frio [ˈfɾiu] (two vowels: “ee‑oo” quickly)

In European Portuguese, use depois de before a noun or infinitive, or just depois as an adverb before a clause. Depois que is a Brazilian pattern and is not standard in Portugal.

• EP: Depois de correr, ela… / Depois da corrida, ela… / Depois, ela…