I've not yet found a way to save images back to the camera in Digicam Control.
There are various options for naming files, but I don't know if you can do anything you want. There's a window on the right headed "session," and the settings menu (gear icon) gives various options for the file name template, as well as one to use the camera's original name. That menu also includes options for where files are saved, including backups, but you can't back up directly to the camera as far as I can see.
Windows sees a connected camera as a device other than a drive. In the backup menu for Digicam Control, you can find the camera, and the files on the card, but when you assign that as a backup, it says the device is busy and it won't allow it. You can, however, designate a camera memory card in a card reader or slot as the backup location, and if you put the card in the camera without modifying the images it can read them.
I find Digicam control works pretty well with the D7100, though I haven't done much with it and haven't explored all its settings. The documentation is lousy, though, and if you get the on line instructions, they appear to be hosted by "manuals online," which tries to sneak other software in if you're not careful. That's a caveat if you download the program itself too. Don't accidentally get the Winzip drivers or some such junk.
e.t.a. I also tried the program "entangle," which does tethered shooting in Linux. It also does not seem to have an option for saving the image on the camera, at least not on a Nikon.