Hello Tyker! The meaning of these sentences does depend on the context, but if you speak about "the cut" it primarily refers to a location that is secluded or hard to find. It can be used in both urban and rural areas. So the sentence; "You can go live on the damn cut as far as I'm concerned." I am guessing that the speaker is saying that the person he is speaking to can basically "get lost" or "disappear". and in the sentence "It's a totally different mindset on the cut." could mean that people living in remote areas have a totally different mind set to people living in busy areas. I hope that helps!

Could they be referring to this place? The Cut is a river in England that rises in North Ascot, Berkshire. It flows for around 14 miles (23 km), through the rural Northern Parishes of Winkfield, Warfield and Binfield in Bracknell Forest on its way down to Bray, where it meets the River Thames just above Queens Eyot Island on the reach below Bray Lock, having been joined by the Maidenhead Waterways. https://en.wikipedia.org/wiki/The_Cut,_Berkshire