If the replacement DVD is for your Mac and not for your sister's Mac, then it will work. Otherwise the replacement DVD will only work with your sister's Mac. Anyway, Mac OS X 10.6 Snow Leopard can be had for 29 USD and it is a full retail version of Mac OS X (excluding iLife though). Handbrake is a multiplatform and open source video transcoder (video converter), available for Mac OS X, Linux and Windows, and totally free. With Handbrake, you can transcode any DVD, video to MP4 or MKV file format with MPEG-4, H.264, or Theora encoded.
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If you need to create a backup of any DVD or convert any video to a format supported by your iPod, one of the best applications to convert your videos is HandBrake. Get used to work with Handbrake, because it is multiplatform, so you'll be able to use it on Mac, Windows and Linux.
You can choose any video format as input. It supports any kind of DVd, no matter if it is protected or not. It also supports VOB and TS. The output formats are a lot: MP4, MKV, AVI or OGM, and it takes advantage of MPEG-4 and H.26 codecs.
Choose the chapters, basic support for subtitles and some styles make Handbrake to be essential if you want to convert DVD into MPG.