If you speak of a router you probably mean a SOHO router like linksys or tplink and so on. They are not just routers, they also share your single wan ip between devices on your network. To do this, they use so-called
NAT[
^] and probably
DHCP[
^] also. These to are not strictly related. There is on side-effect: the inside IP addresses are not visible to the outer world, every site you are visiting is seeing the single WAN IP you got from your provider - that is what you can see with whatsmyip and other similar sites.
A static IP means, that you don't get your IP address from a DHCP server. Your WAN IP and your local IP can be bot static or dynamic, they are not related howsoever.
If you don't use the router you will connect with the WAN IP, but you can't share the connection.
There is one option to have both - but only if the provider and the router is supporting it: pppoe pass-through.
If you got as many WAN IP addresses as many devices you have you don't need the NAT functionality and you can use a regular router instead (or you can switch of NAT, if this option is supported), but than you will need to set WAN IP addresses on the local devices or use WAN side DHCP.