If I fully charge the battery using an external charger overnight, could I use the car for short trips while waiting for the alternator to arrive?
Maybe. I had done that with my VW many years ago...the stock alternator went out (yipee, I get to order the chrome one), so I trickle charged the battery overnight for work the next day.
Did that for two weeks...hard on the battery though.
The VW was a lot simpler electrically, and I avoided as much night driving as possible, as well as leaving the accessories off.
In YOUR case, the engine management systems are electrical, so there is probably more battery draw to be concerned about. Also, as you lose voltage, different components start throwing all sorts of codes to the computer, and your car will probably go into "limp mode".
Unless you have a massive battery like an Optima, I would relegate the car to trips of ten miles or less...