2. even if it were an AP, transmitting at max power for hours, it would give out very limited power (zero, give or take measurement errors).
3. you're starting from assumption "wifi baaaad" without anything to support that claim; if it were a nuclear device, I'd see a cause for concern. [citation-sorely-needed] for this assumption.
Edit: More to the point, it's a good question, even if the answer is "no."
This would be "guilt by association: microwave ovens will cook your brains if inserted, therefore anything else using microwaves will, too, no matter what the power."
Edit: I see that you didn't mean the question rhetorically - so yes, while this may be a valid concern in the design phase, the total exposure is infinitesimal, even in prolonged use.
The electrons flowing through a AA battery are literally the same thing you get out of a 240V appliance outlet, but nobody worries about zapping themselves to death just because the battery makes an electric current.
Edited to add the direct quote.
The answer is "no, your cat won't get its brains cooked, not even after prolonged exposure" - there's not even enough juice in the battery to do that :)
So far the only ways we've found to block the cat also block the wifi.
I'd be significantly more worried(ie: actually worried) about a mishap with that gumstick cell, which looks like it could be(edit: probably is) a lipo.
http://www.hpa.org.uk/webw/HPAweb&Page&HPAwebAutoListName/Pa...
That said, the power levels emitted by wifi devices are so far below any safe exposure limits that it's laughable.
I am not going to say Wifi causes cancer because I don't know (doubt it). But this reflexive bleating about non-ionizing radiation needs to stop. Radio waves do not have to break DNA to affect biological tissues. They can alter many important chemical reactions.
Non-ionizing radiation has biological effects. Magnetic fields have biological effects. This is not controversial!
So?
Also, if you are worried that spending a lot of time "exposed" to a light breeze is going to give you skin disease, you might want to read about hypochondria because you have it.