Subscriptions definitely come down while the Kindle is asleep if the wireless is turned on.
Now. . .it may work differently with the K3 which uses WiFi as the first default. BUT, on my DX. . . I turn it on in the morning and turn on the wireless. . . .then I play for a while on KindleBoards. . .like now. I put it back to sleep while I'm on the computer and when I wake it up a half hour later my daily issues will be there . . . even before the wireless icon indicator has come up to speed and reported the connection.
I think I have had books come too, when I still had my K1. The wireless with that relatively drew a lot of power so, though I'd buy books nearly daily, I'd only turn on the wireless once a week or so when I plugged it in overnight. When I did that, the new content was definitely on the Kindle when I woke it up the next morning.
With the K3. . .if I'm sitting here and buy a book I turn on wireless. . .which is WiFi and talks to my home network. . .and the book is there nearly instantly.
I have also noticed, though, that, if I leave the wireless on and let the K3 fall asleep. . . . books I then buy will be there on the device when I wake it from sleep. It sure seems like they downloaded while it was sleeping, though I suppose it's possible it's so fast that it did it immediately I took it out of sleep. But, again, I feel like they're there before the indicator is fully awake and reporting the connection status. . .but I admit I've not paid close attention.

Anyway. . . .I think the wireless is still active while the Kindle sleeps. . . .but I don't think it's, necessarily, constantly pinging the servers, to see if something needs to come. . .probably pre-set intervals or something -- probably not more often than every few hours and maybe only once a day. And the WiFi may work differently than 3G.
I also suspect that there's a script it runs when you wake it up: I'm awake! Wireless is on! I should check for new content. Which may be why, in some cases, it seems everything downloads when you wake it up.
As to the display. . .no matter what's on the screen, essentially NO power is used to display it. So it doesn't really matter if it's a page of text or an image, or blank. While reading, power is used when you 'turn' the page. Sleep mode does lock the keys. . .which, as basilius suggests, will save power as it doesn't need to monitor whether something is being pressed.