1 When they mean "be about to", they are almost interchangeable. The only difference between them is as follows: If I feel that something will happend in a EXTREMELY short time, I usually say "就要...了". If it will happen in a VERY short time, I usually say "快(要)...了" If it will happen in a short time, I usually say "要...了". For example, Supposing that it's 8.00 now and I'll watch a movie. If the movie will start at 8.01, I usually say "电影就要开始了". If it will start at 8.10, I usually say "电影快(要)开始了". 2 "要...了" can be used to mean "be going to do sth", while the other expressions can't. For instance, 我要去北京了. I'm going to Beijing. 什么时候? When are you going there? 下个月. Next month. 3 "快" can be followed by a particular point in time. For example, "快三点了"(It's nearly 3 o'clock). However, the other expressions can't be used this way.

it is going to rain。天要下雨了 I'm coming home.我就要回家了

Actually, there are almost the same. 意思上基本没有太大的区别，but they have tiny distinct in the point of emphasis. 要...了，就要...了：更强调目的,emphasis is placed on the aim, purpose. 快要...了，快...了：更强调时间上近了,emphasis is placed on the time.可以根据自己要表达的重点进行选择。you can choose by your own expression emphasis.

I think these are so close to each other, they can be exchanged in the sentence. 我认为它们几个可以通用，意思基本一样