Routers use something called 'leases' to assign DHCP addresses. Lease time is the number of days a device keeps its ip address before a refresh. Your router has a setting called lease time. You might try changing lease time to a short period of time. If it is short, change it to several days. Just remember the setting in case it creates problems elsewhere so you can put it back to its original state (or you could just reset the router). Or you could assign it a static IP address if you can identify the slingbox in a router configuration screen. I doubt you have acccess to a configuration screen on the slingbox.
That's an interesting question. Currently the Slingbox does not have a way to refresh to pick up on the new IP address on its own, you would need to manually do so. The following link provides some helpful information regarding how the Slingbox picks up on a new IP address:
Hope this helps!
Sling Media Moderation Team