Is it possible to do this? I'm making a circuit that can be standalone but will also take PC controls. I tried searching but didn't get anywhere.
Yes. It's what Arduino can do when it's connected to the pc over USB, or wirelessly over BT. Data can be exchanged in both directions between an Arduino application (sketch) and a computer program that can communicate with a serial device.
Won't any information transmitted by the arduino to the PC also show up on my LCD?
Check the soft serial libary in the playground. It gives you an extra (but somewhat limited) serial port on two digital pins.
I think the good lady Ada made an even better soft serial lib. but i have not used it myself.
For simply displaying information to an LCD, the software serial library works great. I have been running a 16x2 LCD with it and, I have not had any issues, it is easy to use. From what I gather most of the problem with SoftwareSerial comes from receiving not transmitting.
Yes, i think the most serious problem with the original soft serial is that it doesen't have a SerialAvailable() command. And theres also the max speed (9600) issue.