normally DC socket has two earth pins, should both the earth pins be connected to the earth of the PCB?

 

Many times those pins also serve a mechanical purpose too. So yes I would attach them both

 

what if only pin 3 is connected to the earth of the PCB? could this connection cause trouble?

 

Can you give the part number of the specific connector you are using?

If it has 2 earth pins..connect them both. Companies/people are cheap now.. We don't just put pins there for decoration now.