I'm not sure from your description exactly what software you have with this device. So its a little hard to tell what capabilities it has.
It is relatively easy to cut sections out of your dvd video though. You can use a free program called DVDShrink. In ReAuthor mode you can use the Start/End Frame tool to cut out the exact section of the video you want saved. This will output as a small dvd-video. Read some guides for DVDShrink at www.dvdshrink.info or www.mrbass.org
Once you have the various parts cut out, you would have to reauthor them in the order you want. You'll need an authoring program like DVDStyler to put the pieces into one dvd, with a menu to select each part. DVDStyler is also free to use.
It is also possible to join the pieces together into one file, though without a dedicated editing program, the transitions will be jarring.