There used to a be a method where by you edited the iTunes Music Library.xml file on the second computer to make your device think the two computers are the same computer. I'm not sure if it works any more though. Explanation guide here
Alternatively the easier way of doing it is to take advantage of iTunes file sharing and use an application from the app store. The advantage of this is that most app store video playback applications support more file formats than would be supported though the default video playback app so you won't need to do any conversion before copying the videos across.
The application I would recommend, and use myself, is AVPlayerHD
How do you sync only new songs? You put them in your library, and click sync. If you have duplicates in your library, you will have duplicates on your device. What you see listed in iTunes is exactly what you will have on your device.
The answer is not any harder than this. "Sync" means exactly what it says on the tin, it updates your device with the data in your library. It will "update" songs that have had information changed (cover art, lyrics, or if a title was corrected, etc.), and it will add new ones you've put in the iTunes library.
When you click sync, iTunes does not overwrite everything and start from scratch. It synchronizes your library. It picks up new changes and writes them to the device, and it will even bring new content back from the device if you bought something from the store on your device.
Considering your exact requirements:
Do not change any settings in iTunes. Leave music syncing options completely default, and iTunes will do exactly what you want it to do.
Your exact requirement is exactly what iTunes is supposed to do.
Best Answer
Check this thread out https://askubuntu.com/questions/994/can-i-sync-with-my-iphone-4-and-ipad
otherwise you might ask directly on askubuntu. You can log in on askubuntu using your stackexchange credentials