1. ずいぶん means "very, extremely, surprisingly or considerably".
2. Yes きました comes from 来る。But in this case it doesn't mean "to come". If 来る comes after a ～て form, like (なってきました) in your sentence, it indicates a progressive action that's been going on from the past up until now.
3. Yes. 寒くなってきました means "it's been getting cold".
The translation for your sentence is:
Since it's been considerably getting cold, please be careful not to catch cold.