The best quality will be the original dvd-video. If you convert to something like xvid, or H264, there will always be a slight amount of degradation of the video, though if it is done well, you might never see it.
No video format is more or less likely to corrupt on your hard drive. Corruption can occur, but it won’t be due to the format.
DVD-video can be converted to the various formats used in mobile devices. The question is, do you want your copies already in such a format, or do you want to do this conversion at a later time? The newer formats take less room, and that is certainly an advantage, but as I said, conversion will lessen the quality of the video to a slight degree.
If the conversions are done properly, you won’t see much in the way of glitches or artifacts, but if you compress too much or use too small a bitrate for a particular video, you can see defects in the conversion. Your original dvds may vary in quality of encoding to begin with also, so the amount of compression they can handle without showing defects may vary as well.
Assuming your dvds are not encrypted, you can make an ISO file of the entire disk using ImgBurn. If you wish to play it back, or convert from the ISO, you can mount it in a virtual drive like Slysoft’s VirtualClone Drive. You can also rip to the hard drive as files using most any dvd ripping program. If the dvds are commercially made, and encrypted, get DVDFab HD Decrypter to rip as files. It is the free section within DVDFab and will continue to function even after the trial for the main part of DVDFab expires.
If you prefer a converted format for storage, look into a program called Handbrake. It can give you .mp4 or .mkv files using either the xvid or H264 codecs. There are other choices as well, like MeGui.
All of the programs I have mentioned are free to download and use.