raikyaku no tame ni heya wo souzisimasita.

In Japanese, ため is a noun meaning “sake,” “purpose,” or “reason.” To express “for the purpose of X,” you attach の to X:

• Xのため → “for X’s sake” or “for X’s purpose.”
  So 来客のため = “for the guests’ sake.”

Here ために indicates purpose, not cause. It answers “why did you clean the room?”

• “I cleaned the room in order to (ために) welcome the guests.”
  Contrast with ため (without に) in a causal sense, which is more formal/written:
• 体調不良のため、欠席します (“Due to poor health, I will be absent.”)

No. To express purpose with ため, you must include に. Without it, the phrase is ungrammatical for purpose.
You can, however, use ための when modifying a noun:

• 来客のための掃除 (“cleaning for the guests”)
  But when you attach a verb after ため, you need に:
• 来客のために掃除しました.

Because 掃除する is a transitive verb (a “suru-verb”). You clean something, so the thing being cleaned (部屋) is the direct object and takes を:

• 部屋を掃除しました
  (“I cleaned the room.”)

You can say お掃除をしました or お掃除しました, but it’s more natural to add お to the noun and keep を:

• 部屋をお掃除しました
  Also, you often add お only when speaking politely to someone else, not when referring neutrally to your own action.

Because the speaker is stating a completed action: “I cleaned the room.”

• Past form (ました) = “cleaned (already done).”
• Present/future form (します) would mean “I clean” or “I will clean,” which doesn’t match the context of something already finished.