Automatic download of top apps from iTunes App Store in iTunes
“

Is there a script... to have iTunes download the top (free) apps (in all categories, or specific category) from the app store, like once a week or day?

”
No. But its a really neat idea. And iTunes is very very scriptable.

“

Is there a... other way to have iTunes download the top (free) apps (in all categories, or specific category) from the app store, like once a week or day?

”
Yes, its called AppleScript, and it is the language of iTunes automation.

Unfortunately, it appears the #applescript@freenode.irc channel is quite dead (which says more about irc than AppleScript!), but there are still some great online and offline resources available.
I found a very nice tutorial for beginners to get you started.