Going by the specification of your LG and BenQ drives, both are capable of reading all discs that the Lite-On LVW-1105HC is capable of recording to. This includes DVD+R, DVD+RW, DVD-R and DVD-RW media. Your TMPGEnc 3.0 XPress software is capable of importing recorded content from recorded DVDs and will let you edit out commercials. Just note that your LG drive cannot write to DVDs, however you will be able to use your BenQ DVD writer to write the edited recordings back to DVD.
I have tested 6 different brands of DVD-RW media in my Lite-On LVW-1105HC+ (similar to the model you mention other than the + model is European) and all 6 brands played in all PC drives I have, including DVD-ROM and DVD-writer drives, including a picky LG Hitachi drive.
Unfortunately, I have not used a Sony DVD recorder, so I’m unable to compare its picture quality against that of the Lite-On models. Unfortunately, I can confirm that the picture quality of the LVW-1105HC+ recordings is not as good as its bigger brothers such as the LVW-5045, but definitely much better than what a VCR is capable of recording.