What is good software for video editing under Linux?
There are a lot of answers regarding open source on this answer: Good open source Video Editors?

If you specifically target Linux I'd suggest Lightworks or Kdenlive. Lightworks isn't open source yet but is announced to be after the mac release is finished.

Both a great option for video editing under Linux and fullfill your requirements. Lightworks has only one downside, that being you need to register it on the first start with a free Lightworks account. After that you don't have to connect to the internet again.

If OpenSource is the most important factor I suggest to go with Kdenlive.