double data storage is good
Which one is right when they don't agree?
So, what part(s) of this project are working?
Yes, the Arduino knows when it opens the door, and could tell you that. But, unless you make it remember that it opened the door, it doesn't remember having opened the door. So, you can't ask it later to tell you all about when it opened the door.
As mowcius says, using an ethernet shield frees you from having the Arduino connected to the computer all the time. If you do go with the ethernet shield, you could make the Arduino talk to a server running PHP and MySQL. The PHP script could log, in the MySQL database, when the door was opened.
You could then write another PHP script to generate dynamic html so that you could see when the door was opened from any PC anywhere in the world.
If you can see Twit(ter) on your phone, you can see any other web-based application.
If you don't want to use the ethernet shield, and you are using a Linux-based OS, you could run PHP on the PC that the Arduino talks to, and store the "I opened the door" messages in a MySQL database on the PC or on a web server somewhere, instead.
If you are running windows, PHP can talk to the serial port, but not read from it. So, using PHP on windows is out. But, you could (at least I could) write a C++ or C# application that talked to the serial port and a web server or MySQL database.
we have time and knowledge (we are all beginners) constraints....
Knowledge constraints are meant to be overcome. What are the time constraints? If they are self-imposed, get over it. If they are externally imposed, like homework is due Friday, that's a problem. Better get humping.