Which is more common in Mandarin: 四處 or 周圍?
四處 is better than 周圍 in this case. However, I think you can consider 到處.

The house will have your books everywhere.
這房子將到處都有你的書。

From Google Translate:  


  everywhere 副詞
      到處
      everywhere, throughout, in all places
      隨處
      everywhere, anywhere
      遍
      all over, everywhere
      隨地
      anywhere, everywhere
      普
      everywhere
      廣泛地
      abroad, diffusely, everywhere, large, at large, proverbially
      在在
      everywhere