For this network issues that you are experiencing, it's necessary to determine if trouble is being caused by the network, the Slingbox or the SlingLinks. First, I recommend to hardwired the Slingbox to the router to determine if the issue is with the SlingLinks, just connect a different Audio/video device like a DVD to check if you are able to run the Setup at slingbox.com and connect to the Slingbox without any issues.
If the Slingbox works in that way (hardwired), it will be necessary to check the SlingLinks settings and connections, have a look at this,
Have you tried a different browser like a Firefox? If the issue persists, try Resetting the Slingbox plugin
Is there an order I should be power cycling the equipment in?
Disconnect the Slingbox, router and then the modem from the power, wait a few minutes and connect first the modem, after getting all the lights on, connect the router, wait again for the lights and finally connect the Slingbox. I don't think you need to modify any settings on the router for the SlinkLinks since they are power bridges. You also can try to assign a static IP address to the Slingbox.
Hope this helps