That's a lot of basic questions, which you can find answers for in the various forums around here, but I'll give you a condensed version.
Your problem with your new dvd player seems to be that it doesn't like the media you are using in it. Don't know if it is just incompatibility with the player or if your burner doesn't match up well with those disks. Which model Lite-On do you have? You'll get better answers if we know which burner you are using.
You might be better off using Verbatim brand disks if you buy locally. Or you can try some Taiyo Yuden 8x disks from an online shop like rima.com (if you are in the US)
Are your parents using DVDShrink to reduce the entire dvd? If so, then you might be getting lots of compression artifacts in the copy you are making. Depends on how much compression you are using. Backing up just the main movie and getting rid of extras, the menus, other soundtracks, etc will reduce the amount of compression you have to use. There are many good guides for using Shrink----you might start with the ones at mrbass.org
But compression artifacts probably aren't the cause of the problems you have with the new player.
Since you have AnyDVD you won't run into many dvds that you can't rip. Yes, there are quite a few newer dvds out there with tougher protection schemes, but AnyDVD should be able to handle them if you keep it updated.
If the burner you have now is an older, 4x or 8x model, then you should consider getting a newer one. I like the Pioneer 111 series, but there are good ones from most of the major manufacturers these days. LG and Plextor (if you can afford them) are also good choices.