The old man slept. the window was open.合并成一句话，说明理由

The old man slept with the window open.
with引导的从句在这里做伴随状语.

when the old man slept,the window was open.

The old man slept with the window open.
当两个短句的主语不同时要用with的复合结构，不知对否，做个参考吧

The old man slept left the window opened.
老人没关窗就睡了

While the window was open, the old man slept.

最正确的
The old man slept while the window was open.
while 而，起转折的作用，因为睡觉和窗户的关系是
睡觉前要关好门窗，防盗