I’ve never heard it with ‘girl’ but there is an old fashioned way of referring to the son with the same name as the father by adding ‘boy’. There is a well-known character in a fictional TV series from the rural US of the 1920s called John-Boy. It certainly is nothing I’ve heard in real life and it sounds derogatory to modern ears.

I can help you with English speaking

Hey Aud. The phrase "Kate (being her name I'm assuming) Girl" on it's own doesn't really make sense and is not something someone would say. It could be said whilst being followed by a question "Kate, girl, what are you doing?" But even that would be something a native speaker wouldn't really say. They'd either say "Kate, what are you doing?" or if you didn't know the person "Girl, what are you doing?" Again it's a phrase you would never really use because it almost comes across as rude. Hope that helps :)

I can't say I've heard someone use a name followed by girl, unless he was trying to just be cute? But without seeing it it's hard to guess what he might have meant. In conclusion don't worry about it, it's not normal lol

Yes you can since "Kate" means pure and it's actually a name. So you can use kate to objectify anything related to its meaning.