Sorry its taken a while to get a response, usually people are good about responding. Its wednesday so it makes sense.
Yea, its totally feasible. I personally haven't ever done it (I always check to make sure my door is locked before I leave) but it can be done for sure. If you are looking to send a command to the computer, with the smartphone, while on the same network, it would be a simple matter of getting the computers network address. Although if you are trying to set it up over the internet, then it does get a little more complicated although I know it is possible.
Once you've got that going, then the display shouldn't be too hard. I know freetronics and suretronics makes some led displays that are simple to use. You could also get an lcd from adafruit or sparkfun that would have some good support on getting it to work with the arduino.
If I'm understanding you right, then you would just use the TX and RX pins for the communication between the arduino and a wireless module. The wireless module would be connected to the network, which the computer would be as well which the phone would be accessing through html, I guess. Once you have the wireless all good to go, the display shouldn't be too bad, depends on how nice you want it. You could use a simple 1602 display with an I2C backpack from adafruit which would free up the PWM pins needed to use a door lock servo/motor/whatever you want.