I’m not familiar with any software that you can schedule and play. There are workarounds of course, for instance you could just create a playlist in WMP (or any other media player), but the movies will just play back to back, not at specified times. And I don’t believe they will playback or create a playlist from DVD images, so you may have to convert the movies to Divx/Xvid. But the converted files will take up much less HDD space.
You could do this on about any computer, nothing special is needed. The faster the cpu, the faster your conversions will complete. You don’t need a large amount of RAM (1-2GB for XP, 2GB for Vista) will work fine. About any video card will work, buy you will need a TV capture card if for capturing your movies from VHS.