Tā yuè hē niúnǎi yuè xiǎng shuìjiào.

It’s the correlative comparative pattern: 越 A，(就) 越 B = “the more A, (then) the more B.”

• Here: subject = 她; first clause = 越喝牛奶; second clause = 越想睡觉.
• Both halves share the same subject and each half starts with 越.
• Template: [Subject] 越 + verb/adjective (+ object)，(就) 越 + verb/adjective (+ object).

Most commonly the subject appears once before the first 越: 她越喝牛奶，越想睡觉.

• You can omit the subject if it’s obvious from context: 越喝牛奶，越想睡觉.
• If you front the first 越-clause, keep the subject before the second half: 越喝牛奶，她就越想睡觉.
• Repeating the subject in both halves is possible but usually unnecessary.

Both are fine but slightly different:

• 想睡觉 = “feel like/want to sleep” (desire/urge).
• 困 = “sleepy/drowsy” (physical state).
  So you can also say: 她越喝牛奶越困 (more concise, very natural).

No measure word is needed because it’s a general statement about milk as a mass noun. If you want to quantify, restructure:

• 她喝的牛奶越多，就越想睡觉。
• 她牛奶喝得越多，就越想睡觉。 Using a specific unit (e.g., 一杯) inside the 越…越… frame is less natural unless you have a special context.

Yes, if the thing being drunk is clear from context. You can also front the topic for clarity:

• 她喝牛奶，越喝越想睡觉。 Without context, the bare 越喝越想睡觉 leaves “drink what?” unspecified.

No.

• 越来越 + adj/VP = one thing changing over time: 她越来越想睡觉 (“she’s getting sleepier and sleepier”).
• 越…越… = correlation between two factors: 她越喝牛奶，越想睡觉 (“the more she drinks milk, the more she wants to sleep”).

That sounds awkward. Use one of these instead:

• 越喝牛奶，她就越想睡觉。
• 牛奶喝得越多，她就越想睡觉。
• 她喝的牛奶越多，就越想睡觉。

Use 不 or other negatives in the second half:

• 她越喝牛奶，越不想睡觉。 (less willing to sleep)
• 她越喝牛奶，越睡不着。 (more unable to fall asleep) You can also negate the first half when appropriate.

Yes, 睡觉 is a verb–object compound. In this sentence it stays together: 想睡觉. It can separate in other constructions:

• 睡了一觉 (slept for a while)
• 睡个好觉 (have a good sleep) But don’t say 想睡个觉 unless you mean “feel like taking a nap/some sleep.”