- If I buy the World of Warcraft Battlechest does it include game time, or is that only sold separate?
You'll get a free 30 days when you upgrade your trial account.
- Do I also need to get the Wrath of the Lich King to play?
No, but you won't be able to get past level 70, get into Northrend, or create a Death Knight. You need to have a character at level 55 or higher in order to create a Death Knight, so you can wait until then to buy Wrath of the Lich King without missing anything.
-Will I be able to keep the characters I made on the 10 Day trial?
Yup, just make sure to upgrade your trial account on Blizzard's account management page instead of creating a new account with the CD key that comes with the game.
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Remember: with great power comes great current squared times resistance.