Technically, the NMEA standard fixes the baud rate at 9600, but I think most GPS receivers allow higher-rate connections. I use the iBlue receiver, and its specs say "up to baud rate: 115200 bps". I have no idea what baud rate it is actually using, though.

I have seen the BT drop, but only very rarely. I have seen it go down after 45 minutes, and I have seen it last for an entire 2-hour trip. I have never seen a pattern.

Personally, I highly doubt a baud rate higher than 9600 is really going to buy you anything in terms of functionality, because the data that is being pushed is almost definitely not enough to saturate a 9600-bps pipe.