Anytime you increase the magnification of an image (as you do when you zoom in on your computer monitor) you will increase the perception of noise. You will also notice details that are ever so slightly out of focus, even though they are within the depth of field (DoF) for a given focal length, aperture, sensor/film size, print size, and viewing distance. In general, DoF is figured based on an 8X10 viewing size at 10" (25cm) by a person with 20/20 vision. Noise works similarly.
One way to reduce the noise in your images when using high ISO is to save the images as RAW files and adjust the Noise Reduction (NR) settings when converting the images to JPEG on your computer. This gives you much more control over the relationship between noise and detail sharpness in each image. When you save your in-camera files as JPEGs, you allow the camera to use a "one size fits all" approach for that particular ISO. This may or may not be the optimal amount of NR for that particular image.
Based on several independent reviews of the D5200 and comparing the scores from the same reviewers for the cameras I use, I would say somewhere around ISO 3200 should give you excellent low noise performance for producing 4X6 prints. By ISO 6400 at one stop faster, when viewing the prints you will probably be able to detect some noise in dark areas that are not totally black.