Yes, you can connect both ethernet ports and they will speed up your internet access.
No, it will not double your speed; at least that is my experience with a dual ethernet setup.
Internet access speed is dependent on several things: the number of computers making requests on your local LAN (if you have cable or FIOS access), the sever speed(s), packet arbitration, etc. By making simultaneous dual requests the servers recognize it as a single request (otherwise you would be getting double data downloads for each request) and will send the data to the fastest receiver. You will have the advantage of having two places in line, but the limitation is your request can only be filed once. So you will have increased speed, but not double speed.
To monitor your internet speed I suggest speedtest.net.