Ethernet ports on wireless access point

Yes, the other LAN ports will work -- and when you attach the AP to the router you will want to use an LAN port at each end of the connection.

To set up the wireless on the AP, turn off DHCP, assign the AP either a reserved dynamic address or static address outside the DHCP range but in the network, use the same security type and pass key, and use a different set of radio channels from the router.