DX v 1.3 mode

Is there any advantage in using the 1.3 crop mode on the D7100 or is it basically just the same as using the DX mode and then cropping.

Since the 1.3 image setting uses only a portion of the sensor it is virtually the same as cropping to the same size. The only advantage is you can shoot more image per second in the 1.3 mode. The advantage to cropping is you can also change the framing at the same time.

The only advantage of doing it is reducing the file size, which couldbe an advantage on a camera with a buffer as small as the D7100 which I believe clogs up after 6 raw images in Continuous High mode. You may only get one more raw frame, but you can likely up your shots per buffer fill by 30% shooting JPEG. If you're doing events and need to send unprocessed JPEGs out immediately then it could also help you in situations where you don't have enough reach.

Off the top of my head, I recall needing to use this mode to use higher frame rates for movies.
I find these comments on the internet:

When shooting in this mode, the resolution is reduced to 15.4 million pixels, but it does mean that the 51 autofocus points fill the entire frame and Full HD movies can be recorded in 60i and 50i.