Change Wired Router IP address

Using a Dlink wired router for 2 Mac Mini's.

works fine, but need to understand more about the IP address as having problems with a specific host server and changing the IP address should solve it

Comcast provides the connection. Do they issue the IP #? If so, is it likely they can change it?

Or is there a way that restarting the router will cause it to automatically have a different number?

Another test option: at one time each mac mini was connected to the internet without the router. One at this physical location via Comcast. The other at a different physical location with a different internet provider. If the router is temporarily taken out of the connection to the Comcast internet connection, and one of them is directly attached to the Comcast Cable will their 'old' IP address be the one that is in use?

(Having a different IP will help show a hosting firm the problem is at their end (automatically rejecting a connect to a specific server, when it is possible to connect to a different server on their site which is at MacHighway). For 10 days they've been saying it's my computer that is faulty, but I can FTP the exact same material created in Freeway Express 6.11 to a different MacHighway computer via a different user name and password.... Yet, they keep saying they have "turned off the stop" that prevents the connect - but it never connects from the FTP attempt for it is automatically re-stopping the connect as shown with a FileZilla test). I calculate that changing the IP (that specific server is rejecting automatically) will finally get through to them to assign the hosting account to a different server - which I've proven I can connect into.)