Let me answer the last question first.
Your computer is less vulnerable if you have a router because by default all the routers are acting as firewall.
If you are using DSL connection then your routers external IP address will change every time you power it down but if you are using Cable broadband then you IP address would tend to stay the same (for a long time). IP addresses are handed out to the router by your ISP's DHCP server and, usually all the DHCP server have a lease time for IP addresses (usually 72 Hours)... Here is the what makes the DHCP server give your router a same IP address... they call it MAC address...EVERY network device (Router, PCMCIA, PC network card) has a unique MAC address that means there aren'ts two device in the world having the same REAL STATIC IP Address from their ISP (please do not confuse it with Internal IP address).
There is a lot more into the DHCP server then what I have mention
http://www.dhcp-handbook.com/dhcp_faq.html
Cheers