The label 'lady in red' originated with a woman named Anna Sage who assisted the FBI in the capture of gangster John Dillinger in 1934. The oldest profession is prostitution and typically those women are called 'ladies of the night' but Anna became the 'lady in red' due to her outfit she was wearing the night Dillinger was caught, she was also a Madam of a brothel. Her outfit was actually an orange skirt and white top but the nickname was given nonetheless based off of her promise to wear red - even though she changed her mind about that. You can read all about it here .  How it came to be a saying today and what it actually means is a mystery but there is a song about it, sung by Chris De Burgh. 

In the red comes from accounting practices. It seems that red ink was used to note a deficit while black ink signaled a profit. ( source )  Saying the phrase today means that we are also experiencing a deficit. 

Being caught red-handed was a way to catch poachers back in the day, the red referring to the blood still on their hands from killing animals. ( source ) There seems to be some other ideas to the origin but this one seems to be the most plausible.   Today the phrase means being caught in the very act, for example the below picture of me when I was a wee one. 😏 When I was a wee one I got into my Grandma's powder puff and used it liberally on myself and covered the bathroom with the stuff. I was caught red-handed and a picture was taken because I was just too cute to scold! :) 

Red herring was used to divert or distract bloodhounds from their search for fox. People who were against the sport would use it to mislead. ( source ) Today it is used to indicate a diversion of any kind.