Я відкриваю шафу і вибираю одяг.

Because they describe ongoing or habitual actions. In Ukrainian:
• Imperfective verbs (infinitives ending in -вати) express processes rather than single, completed events.
• Ukrainian does not have a separate “-ing” progressive form; the imperfective present covers both simple-present and ongoing actions.

Use the perfective forms:
• Past: Я відкри́в(а) ша́фу
• Future: Я відкри́ю ша́фу
For “choose” you get:
• Past: Я ви́брав(а) одя́г
• Future: Я ви́беру́ одя́г

одяг is a mass (uncountable) noun in Ukrainian.
• It does not have a plural form.
• Its accusative form is identical to the nominative, so you say одяг in both cases.

Yes. Ukrainian is a pro-drop language. You can simply say:
Відкриваю шафу і вибираю одяг.

– і is the neutral “and.”
– й replaces і only before a word starting with a vowel (e.g., й одяг, but not й шафа).
– та is a colloquial alternative (e.g., …та вибираю одяг).
All three can link clauses, with slight tone differences.

Both mean “to choose,” but:
• вибирати is used for picking items (clothes, food, etc.).
• обирати often means “to elect” or “to select among candidates,” though informally it may be used like вибирати.

Insert a possessive pronoun in the accusative, feminine singular:
Я відкриваю свою шафу і вибираю одяг.

Yes:
• гардероб – “wardrobe” (often built-in or walk-in)
• комод(а) – “chest of drawers”
• пенал – “cabinet” (e.g., for books)

• ша́фу (stress on ша́)
• відкрива́ю (on ва́)
• вибира́ю (on ра́)
(Regional variations exist; consult a good dictionary.)