Can I password protect/childproof iPad apps? I have ordered an iPad 2. I was wondering if I could protect applications that I don't want my 7-year old to muck about in.

It's stuff like Mail, Things, Evernote, and perhaps some games not suited for a small child.  

I'm OK with paying for it if there are good third party solutions but it should be able to protect any app.
You can't generally password protect specific apps without some jailbreaking I assume, but you can restrict certain things, a few of which I recommend if you are giving it to a young one.
http://support.apple.com/kb/ht4213 Describes the restrictions process, and it lists

You can choose to restrict access to the following applications and features on the device:
Safari,
YouTube,
iTunes,
Ping,
App installation,
App deletion,
Camera,
FaceTime,
Location,
Account changes,
Multiplayer games in Game Center,
Adding friends in Game Center.
In addition, you can choose to prevent access to specific content types:
In-app purchases,
Music and podcasts,
Movies,
TV shows,
Apps.

You can also set iTunes restrictions for content via http://support.apple.com/kb/ht1904, but again, this isn't device specific at the time, it is when you are buying or downloading content from the internet.