My Hp laptop won't power anything when it is in standby, so that would be a no.
They say a good mechanic only needs 2 tools - WD40 & Duct tape. If it moves, and its not supposed to, duct tape. If it doesn't move, and its supposed to, WD40.
Yeah, most laptops won't power the USB port when they sleep or are turned off. Some of the newer ones have a cool "Sleep-n-Charge" feature that will power one USB port while the computer sleeps. Definitely not a feature to buy a new laptop for tho.
Just about any USB charging wall wart should work with it. I found that my iPod doesn't like my Chinese brand USB charger, however, so I'm not sure if the transmitter cares what it is plugged into.
And what's wrong with leaving the laptop on to charge the transmitter? Just close the lid and it won't use hardly any extra power if no programs are running.