最近天气慢慢越来越冷了，今天早上我不想离开被窝！

慢慢, 越来越.... are similar words as adv.； meaning gradually or incrementally in English
So you can write ;
1] 最近天气越来越冷了
2] 最近天气慢慢变冷

昨天晚上睡得不舒服，好，冻醒了好几次。

不舒服; in a small range, sound like something effect you don't sleep with, it is hard to explain. en...........
睡得不好 could describe in a big range.


I add 冻 for a relationship with the last sentence. because the weather is cold, I am wake up a few at night by cool.

六点钟起床，急急忙忙准备匆忙洗漱，六点半就出门了。

急急忙忙 和 匆忙 are similar words. it doesn't have a huge difference to me as Chinese readers.
for the last part, you can keep the " 了“ which means the tense is pass and finish.

幸运地的是，今天电车没晚点，能准时上班了。

早起累死了。。。......

I have a different idea with your last sentence. Jack changed one-word " 今天” which means “ Today, I am so tired."
but in your original sentence, your means; " wake up early is tired."
so let me know what ideas you want to say.

最近天气慢慢越来越冷了，今天早上我都不想离开被窝起床！

昨天晚上睡得不舒服眠不太好，醒了好几次。

早上六点钟起床，急急忙忙准备洗漱完毕，六点半出门了。

幸运地的是今天电车没晚点，能准时上班了。

今天我不怎么忙，学习一下，跟同事聊聊天了。

早起今天累死了。。。