As others have stated it must be Lightroom which is highlighting blown areas with red.
Before you condemn the images, open them in View NX and see (and show us) how they look. Just load the images into View NX and export them as jpeg without any processing. Then post them in this forum.
The D5200 has enough DR, so you can meter in the bright area (use matrix metering) and shoot in aperture priority. Once loaded in View NX, you can lighten the dark areas selectively by using Active D lighting presets. Just remember if it is too bright then the sensor saturates and you loose data (you cannot darken the scene and get details). If it is dark, the data is there and you can brighten the scene and see the details.
I have not much idea of the D5200, but in D3300 and other bodies I have, you can view the histogram in a number of formats - Intensity, Red, Blue, Green or all of them. Individual colours can saturate the sensor and blow the highlight, even though the combination has not, so it is wise to check individual colours in the histogram.