1.这个句子有错,如果选择 whoever,应该修改为 whoever leaves the room last out should lock the door.
Whoever = Anyone who 或 He who； 既是连词也是代词,如修改后的本句：
= anyone who leaves the room last out should lock the door.
句子中 whoever 既引导了主语从句（= 可以化为定语从句 anyone who leaves the room last out ）,又兼作should lock the door 的主语（= anyone/he should lock the door）
另如：
Whoever comes will be welcomed.无论谁来都会受到欢迎.
本句中的 whoever 既引导了主语从句（= 化为定语从句 anyone who comes）,又兼作will be welcomed 的主语（= anyone/he will be welcomed）
2.The Parkers bought a new house but ▁▁ would need a lot of work before they can move in.
1）but 后的句子中省略了不定式 to do 恢复出来便一目了然:
it would need a lot of work to do before they can move in.（在他们搬进来之前还有大量的工作要做）
可见 it 是形式主语,被省略的不定式 to do 是真正的主语 .
2）另外 need 指的是需要必需之物,The Parkers would need a lot of work 意思就成了“帕克一家需要找到大量的工作”,言外之意是可能他们是想赚钱或者是没事找事干.