Do I need to completely finish Season 1 before starting Season 2?

Yes the second season of the walking dead is a direct continuation of the story from the end of the first season and I would imagine there will be some large spoilers from the end of season one that will become apparent almost immediately.

My advice is finish season one first.

I am not sure how the 400 days DLC ties into season 2, if at all. Maybe some one else could add that information.

You don't "have" to but I think it would not be a wise decision. They allow you to select a pre-set selection of "Decisions" if you choose not to play the first, but it is definitely not recommended.