If you physically power off your router, and turn off the wall plug switch, it will not magically power on. That much I'm sure 

Wether Maxis will send a technician to complain and force you to plug it in, I cannot answer. Please do join in the forum thread I posted above your post and pose these penetrating questions to them.

However, if you leave it plugged in but turn it of... I don't know if it's possible to force it to wake up. For normal PCs however, I used to work on a computer that could do that if my admin forced it to wake up, back in uni.
As long as it was plugged into the wall and the switch was on (but PC is off). Was pretty creepy to see it suddenly spin up to life.

hi, Yes mine is the 784n:

Product Name:  TG784n
Software Release:  8.4.H.8

I must be one of those first batch subscribers who installed M's FTTH when they first launched it back in near end of 2010.  To be honest, I had have wifi problem when I realised more n more neighbours did the installation... As I suspected the channel conflicts, changing the wifi channel to one that hasn't been allocated helped eradicate the disconnection issue.. And I have no issue until now.  The only issue that I have with this 784n is that it doesn't save my devices' configuration in the 784n, and this is due to Mshits keep on pushing a 'refresh' back to the router I believe.. 

cheers

p.s : that was wifi connection problem that I have had.. It has never overheated, not as bad as you mentioned...every equipment will generate heat, and somehow 784n last 18 mths of almost running 24*7 inside my open-cabinet..  so cheer up!