tā zǒngshì bǎ kěyǐ huíshōu de lājī hé bù néng huíshōu de lājī fàng zài bùtóng de lājītǒng lǐ.

把 introduces a “disposal” structure: subject + 把 + object + verb + result/place. It emphasizes what happens to the object (the trash).

• With 把: 她总是把垃圾放在不同的垃圾桶里。 – Focuses on what she does to the trash.
• Without 把: 她总是在不同的垃圾桶里放垃圾。 – Still correct, but the structure is more neutral; the emphasis shifts slightly to where she puts trash rather than disposing of the trash as a whole.

The main verb is 放 (“to put, to place”).
The core structure is:

• 她 (subject)
• 总是 (adverb of frequency)
• 把 可以回收的垃圾 和 不能回收的垃圾 (two objects handled by 把)
• 放在 不同的垃圾桶里 (verb + prepositional phrase of location).
  So: 她 + 总是 + 把 + [trash A] 和 [trash B] + 放在 + [place].

总是 means “always” and is an adverb of frequency. In typical word order, it comes after the subject and before the verb or 把:

• 她总是把垃圾分开。
  Moving it to other positions (e.g. 她把垃圾总是分开) is usually unnatural or wrong. Stick with subject + 总是 + verb/把.

Here 的 turns the preceding part (可以回收 / 不能回收) into an adjectival clause that modifies the noun 垃圾.

• 可以回收的垃圾 = “trash that can be recycled / recyclable trash”
• 不能回收的垃圾 = “trash that cannot be recycled / non-recyclable trash”
  So 的 is acting like a linker for “(that is) …” before a noun.

In Chinese, modifiers generally come before the noun. Relative clauses like “that can be recycled” are also placed before the noun they describe.
So English “trash that can be recycled” becomes:

• [可以回收的] 垃圾 = “[can-be-recycled de] trash”.
  The entire 可以回收的 chunk functions as an adjective attached to 垃圾.

As written, each part is a full parallel phrase: 可以回收的垃圾 and 不能回收的垃圾. The repetition makes the structure clear.
You can shorten it in natural speech or writing:

• 可以回收的和不能回收的垃圾 (omit the second 垃圾)
• Or even 可回收和不可回收的垃圾.
  But you cannot drop 的 while keeping the noun: ✗ 可以回收垃圾和不能回收垃圾 sounds off because the modifier-noun boundary becomes unclear.

在…里 expresses “in / inside …”.

• 在不同的垃圾桶里 strongly highlights the inside of the bins: “in different garbage cans”.
• 在不同的垃圾桶 is sometimes used in speech, but it sounds less complete and can feel more like “at/at the location of different garbage cans”.
  Adding 里 makes the spatial relationship clearer and more idiomatic: inside the bins.

和 here means “and”, joining two parallel noun phrases:

• 可以回收的垃圾 和 不能回收的垃圾.
  You can usually replace 和 with 跟 in casual spoken Mandarin with no change in meaning:
• 可以回收的垃圾跟不能回收的垃圾.
  和 is slightly more neutral/formal; 跟 is often more colloquial.

Yes, common alternatives include:

• 垃圾箱 – trash box/bin
• 垃圾筒 – trash can (similar to 桶)
  In most everyday contexts, 垃圾桶, 垃圾箱, and 垃圾筒 can all mean “trash can / garbage bin”. Swapping them in this sentence does not change the basic meaning, though some regions or contexts may prefer one term over another.