No need to buy an external until you try the one that is built in. There is no telling at this point what manufacturer Dell is using for that laptop drive, but give it a shot.
I’d buy Verbatim 16x media…if you buy locally, look for the AZO designation on the packaging. Verbatim has been changing to cheaper sources lately, but their original disks using the AZO type dye are still available at many locations. Your drive has a top speed of 8x, but these faster disks will still work with it. + or -R doesn’t really matter much anymore, but I wouldn’t use RW unless you think you need to experiment a little and want to use disks that can be erased. If you do decide to use RW disks, buy +RW Verbatim.
Printable disks are not prone to degradation any more than the shiny top ones. So, what you read is not true.
Taiyo Yuden bought JVC media division. So, the premium line of JVC disks are now made by Taiyo Yuden. Results from them have been mixed so far. Which is why I would recommend buying Taiyo Yuden 8x +R disks from rima.com or supermediastore.com if you plan on ordering disks online.
I haven’t tried to copy from a DVR to a computer, so I am quite leery of trying to advise you on that. I have used several different capture cards in the past to capture straight from tv feeds, from cable and satellite, but not from a DVR. Hauppauge is a good brand for capture cards, but they may not have anything appropriate in your price range.