She likes to have her room look neat and tidy？

先看 have 在这里的用法，如下图所示：

上图所示的都是 have sth./sb. done 的用法，而你给的是 have sth. do 的用法。

She likes to have her room look neat and tidy？

把上句中her room look neat and tidy 从新组合为一个句子如下：

Her room looks neat and tidy. 这句是主系表结构中特殊的一种现象， 本质上是

Her room is neat and tidy.

所以上句：She likes to have her room look neat and tidy 中的look可以理解为 is，相当于：

She likes to have her room neat and tidy。

更多have 用法请参考百度：

have sth done的四种用法