I just wonder, finally if I finished the soldering the circuit. Do I need to install any firmware (Like BIOS in PC) to set up?
It depends on your setup. How will you program the '328? ICSP?
As I found in original eagle file of arduino mini, it includes a crystal with four pins(x2 16MHz), but the pads are under the crystal, so that it is very difficult to solering. As a result, my supervisor suggest me to change into another package, but recently i can only find one (package hc49s) in stock, but it has two pins. That means I am not able to connect the crystal into ground.
That would be a Crystal. The original Pro Mini uses a Resonator. If you are going use a Crystal, you will need 22pF caps on each leg connected to GND.
Will it effect my testing if I do not connect the crystal into GND ？
Yes, it won't work..