Hi,
yes, the driver's footwell compartment is quite big and is normally empty apart from the rear speaker amp. Cars with the boot mounted sub-woofer (BOSE) however have the toolkit relocated under there but still have enough room for the larger amp.
With the standard arrangement the head unit's own internal amp runs the front speakers and the secondary one in the footwell running the rear. The rear line out signals go from the head unit to the amp and the speaker feed from the amp to the rear speakers.
Cars with the BOSE sound system have a larger amp located in the same place which runs all speakers. The head units amp is not used and all 4 line outs are run to the secondary amp. All Audi head units can be programmed to select whether line outs and/or the internal amp is to be used.
To upgrade the system you could replace the existing rear amp, but the front speakers would still be run from the head unit. To run all 4 speakers sets from an external amp you would need to disconnect the front speaker wires from the head unit and run them together with the front line outs from the central console down to the foot well. All Audi head units have a sticker on them which shows the pin outs which helps work out what's what. I've attached a picture of the pin-out sticker and head unit wires from a car with the rear amp for reference.
If you are increasing power output you may also want to beef up the feed to the amp. For example, the cars with the more powerful BOSE amp have a thicker wired main power feed plus and additional secondary feed from an extra fuse over the standard smaller amp.
The rear only amp is simple in operation. There's the power feed, run from fuse 2 (the same one which runs the head unit), line ins for the two channels (4 wires) and outputs for the 2 speaker sets (4 wires). The last one is a +ve power on signal from the head unit with turns the amp on when the head unit is turned on.
Pins:
2 red/white +ve feed from fuse 2
3 brown -ve feed
9 white rear left line in signal (from pin 1 on head unit)
4 brown/white line in ground (from pin 3 on head unit)
10 yellow rear right line in signal (from pin 2 on head unit)
5 red/blue power on signal (from pin 6 on head unit)
7 red/green left speaker +ve
1 brown/black left speaker -ve
6 blue/black right speaker +ve
12 brown/white right speaker -ve
regards
Andrew