我的女儿累死了

How about:

女儿同学的父亲昨天出了点车祸。尽管他没什么事，他的孩子们（双胞胎）却睡在了我家。孩子们很兴奋，只睡了一小会儿。他们六点就起床了。今天是星期六，女儿有游泳课，她今天真是累坏了。

A full stop represents the stop of one logic thinking. If the two sentences are linked logically, please use commas in Chinese. I hope it is useful to you.

Following last correction.

睡觉时间很少 is grammatically correct, but a more natural and colloquial expression would be 睡得很少. 

他们今天6点己经起床了。Nice!

今天是星期六，女儿去上游泳课。她累死了.-累坏了

去上xx课  sounds better, although sometimes 我去xx课 is acceptable and 上 can be omitted..

I feel that it is a very nice entry. You got all the structures correct, but just some small mistakes on word usage. 

我的女儿累壞了

自"己" oneself vs. "已"經already: the difference between the Chinese characters is that the last strike "已" in 已經 is longer than "己" in 自己 to project from the level of the second strike.

我的女儿累死了

我的女儿累死了---累坏了 sounds better in this case, because 累死了 indicates a much stronger level of tiredness.

女儿的同学的父亲昨天有交通事故。女儿同学的父亲昨天出了交通事故。

女儿的同学的父亲 sounds wordy. When you want to use two 的 in one adjective structure, be careful, review the sentence and see whether there is one 的 can be omitted. 

有交通事故 is used to describe "an accident happened in a certain place", e.g. 这条街昨天有交通事故/这条街昨天出交通事故了。When describing "a person had an accident", use "出事故". 

虽然他没事但是孩子们(两孪生)在我们的家里睡觉了。虽然他没事，但是孩子们（一对双胞胎）在我们家睡觉。

Always remember to use comma to separate a sentence..孪生 is okay, but the more common expression is 双胞胎. 我们的家里 can be shortened to 我们家.

但是他们很兴奋的。睡觉很小时间。但是他们（都）很兴奋，只睡了一小伙/睡得时间很少.

睡觉很小时间 is incorrect. First, 小 can be used to describe time, but 时间长/短 or 多/少 can do. Second 睡觉很少时间 is not correct, we usually say 睡得很少/只睡了一小伙.