Eu separo a roupa limpa da suja na gaveta.

Separo is the first‐person singular in the present indicative of the verb separar. It means “I separate” or “I am separating”, usually expressing a habitual or general action rather than a one‐time event.

In European Portuguese, descriptive adjectives most often follow the noun:

• a casa branca – “the white house”
• a roupa limpa – “the clean clothes”
  Putting an adjective before the noun (e.g. a branca casa) is possible for stylistic or emphatic reasons but is less common in everyday speech.

Da is a contraction of de + a. After separar, we use de to indicate what you’re separating from:

• separar A de B
  Thus da suja = de a suja = “from the dirty one.” Here, suja stands for roupa suja (“dirty clothes”).

Yes. Portuguese is a pro-drop (null-subject) language. You can say:

• Separo a roupa limpa da suja na gaveta.
  The verb ending -o already tells you it’s first‐person singular. Including Eu adds clarity or emphasis, but it’s grammatically optional.

Na is the contraction of em + a, meaning “in the” (feminine singular). You use em to indicate location:

• em a gaveta → na gaveta (“in the drawer”).

To split or sort things, Portuguese uses separar X de Y (“separate X from Y”).

• separar A de B
  You don’t say separar entre…, because entre means “between” in a spatial or metaphorical sense, not in the sense of dividing or sorting.

Yes, grammatically it’s correct. The only difference is location:

• gaveta (drawer) is typical for socks, underwear.
• armário (wardrobe/closet) is typical for hanging shirts, coats, etc.
  Choose the noun that matches where you actually store the clothes.

Roupa is a feminine noun. Adjectives in Portuguese must agree in gender (and number) with the noun they modify:

• roupa limpa (feminine singular + feminine singular)
  If you talked about os casacos (masculine plural), you’d say casacos limpos.

In European Portuguese:

• s before a vowel at the beginning of a word is pronounced [s].
• j is pronounced [ʒ] (like the “s” in English “vision”).
• final a is an unstressed [ɐ] (almost a schwa).
  So suja sounds like [ˈsu.ʒɐ].