DPP applies "capture sharpening"/"input sharpening" to the Raw data the same as the camera would apply if you are shooting jpegs.
Whether or not an image "needs" another sharpening process is largely up to personal preference, with consideration for the specific image and the desired output.
For a lot of shots, ones that I want to make a quick conversion and post to my Web host, I don't typically fuss with going beyond the input sharpening. If it looks good viewing at a "normal" viewing size, I'm content.
But if say I'm preparing a shot to print at a large "framing" size, well, I'll give more thought and effort to sharpening considerations. If, say, I'm preparing an image to print at my typical 12x18 size, I'll look at the image in it's "print size", which will fill my screen (horizontally) and that way I can get a look at what it might need as far as sharpening. If I need to go farther than the Raw, say for resizing and resampling the image to get a resolution of 300ppi, then I'll do that first before "output sharpening".
Another consideration for certain shots is "selective sharpening". For example, if I have a shot of a critter I'll often want a "final output" to have sharpening applied only to the critter, with the background remaining softer. DPP can't do this, but an image editor with layers and masks can make this operation fairly straightforward. Lightroom makes this doable as well.
If I do need to go into an image editor I'd typically convert the image to a tiff for this to avoid repetitive jpeg compression. In DPP you can use the Tools/Transfer to Photoshop if you have Photoshop.
If you don't have an image editor, well, get one -- Photoshop Elements is capable of doing all types of tasks at an affordable price.
For DPP only, you can't do the selective sharpening, but you could Convert and Save as a tiff, you can resize to your intended final use, and then apply sharpening at your intended viewing size. As you've noted, once you have converted from your Raw format, you are limited to your RGB controls and are beyond "input sharpening". The Sharpening control in the RGB tab will add to the effects of the input sharpening in the Raw tab.
Again, your output sharpening should be use/view purposes. Input sharpening is best viewed at a 100% view. It may (or may not) affect the shot viewed at the "normal" size. Again, the normal view is best for output sharpening.