Yes, you can put up to 7.9gb of data on a DL +R disk. There should be no degradation of quality when burning to a DL disk, though if you are burning dvd-video, I would suggest using ImgBurn as your burning program. It will set the layer break correctly, while some burning programs will not. ImgBurn is free to download and use.
From the brief bit of research I did on your drive, it appears to be a rebadged Lite-on 20x model. I have to assume it will burn up to 8x with DL media, but Lite-on’s are not known as the very best at burning DL disks, so I would stick to 4x speed. Verbatim +R DL disks come in two varieties, 2.4x-6x and the 8x disks.