A bit of help
Its been some time since I have edited a DVD instead of from a tape or from a camera (won't get into all those formats since its useless at the moment).
I'm assuming the DVDs were made into a "normal" DVD format which does cause some problems when moving them into iMovie. The reason being iMovie will think its like a studio DVD. So handbreak may be an option, though I like Cinimatize, it's an older program but works for me. The main thing is getting your DVDs into QuickTime files, just a warning making them into a QuickTime file can take up a lot of hard drive space (there maybe another program that gets suggested to work better, and use less drive space). From my expierence QuickTime files work great, granted I started doing this pre handbreak.
I would try handbreak first since its free and hear from others before buying anything. Once the files are ready for iMovie then just import them into iMovie. You shouldn't loose any quality.
I can answer your second question with ease. Yes iMovie will make a new DVD, well technically it will send it to iDVD where you make the menus, other little things you may or may not care about. Not sure what size DVDs you use but most home videos can get on your typical 4.7GB, though you may need to get a larger size depending on how long the video is, and if your DVD drive can burn dual layers, most new apples can.
Yes the whole thing can be done with ease after the first disc, the first is the learning curve, not that it's much to learn, just an easy program then iDVD (guessing you already know iMovie, if not imovie is pretty simplr, after about an hour of messing around with it you should know it well)
Let us know the format of the DVDs, again sounds like a simple DVD we all make, kinda like mine after I use Toast. I hope you get more replies that are simple. There may be a easier way to do this since there shouldn't be any copy protection, again my way is kinda old by about 5 years ....except , you should have it no matter what, great application. Also people on here can help if you get stuck along the way. Good luck, hope to hear something you tried works since editing can be fun...to us sick and twisted editors.
I would also pick up MacTheRipper or ripit (Mactheripper is free) to have a backup of your home DVDs, and Toast or Popcorn if it's in your budget, then you don't need to worry about doing the whole process of getting home movies to DVDs again.