Let me help you with this.
The processor in mentioned works at 77W but there is another version (low power consumption) model Intel® Core i5-3550S that runs with 65W.
You can check this information at the following link:
Based on the information Gigabyte provides about processor compatibility, it seems your motherboard should run this processor. Here you have the processor compatibility list from the Computer Manufacture:
In the case the system is not working fine I would recommend you checking all connections on the motherboard and if possible testing a different processor. By doing this we can discard issues either the processor or the motherboard.
Just noticed something on the MOBO site for this board. It says some Intel i5 processors require a video card. Do I need the Video Card installed before running BIOS for 1st time? I had the monitor connected to the MOBO itself,not the graphics card.
This may have been a typo but I can confirm this is definitely a 77W product.
In regards the graphics question, this processor comes with onboard graphics so there is no need for you to get a video card since the motherboard will use the onboard resources.