That's right DNG is an open source "Raw" format maintained by Adobe.
The thinking is that Raw formats come and go and constantly change but as long as there is an Adobe there will be a standard, a DNG raw format.
That's why you can't save as a DNG in Photoshop, you've just converted from one raw format to another, anything you save in Photoshop is an image file not a RAW data file.
You've save a "copy" of the original ARW as a DNG on your computer, but you still have the ARW file open and active. If you click on open image, you're opening the ARW file..
No you're not worrying on the raw file any more, actually you never worked "on" the raw file.
Any changes you made in Adobe Camera Raw were data instructions stored in the xmp file. The raw file is the same "data file" that came out of the camera.
IMHO, it's not a good or bad idea to convert your raw file to DNG, but if you do, and you're positively sure they're OK and not corrupt, delete the ARW's.
You can batch convert your raws to dng with the Adobe DNG Converter .
I don't convert them to DNG and I never, ever delete my raw files.
What I do is something like this.
I shoot only RAW and open the file in ACR.
Since any changes done in ACR are non destructive I do as much as possible including white balance, cropping, straitening, etc.
I finish up in PS and save as a PSD file.
It's compressed but lossless.