My research shows that when you unplug a router it will automatically assign a new IP address, I am not understanding at this time why I would need to change an IP address myself in help desk remote network troubleshooting?
What happens when you restart a router/modem/CPE (customer premises equipment, general term for equipment that connects the customer to the ISP) depends on the ISP. Some ISPs will assign a new address to each session/lease, some will remember the last assigned address for some variable time and assign that same address, and some assign static addresses (customer has one specific IP address tied to him). The goal usually isn't to actually change the IP address, but to obtain a new lease/start a new session (depends on the technology used by the ISP). A new IP address will be a confirmation that this succeeded, but in a customer support scenario you should be able to look up the actual sessions/leases, and not rely on a changed IP address.
About default gateways (and an opportunity to learn about subnet masks): Each network device has an IP address, and also a subnet mask. The subnet mask tells the device which addresses are local, and which are not. Local addresses can be reached directly on the network. If the device wants to connect to an IP address that is not local, it will check its routing table (see "route print" in Windows command line) and figure out if there is a defined route for the network the remote IP address belongs to. If there is a route in the table, the device will follow it. If there isn't, it will go through the default gateway. In other words, if the address is not local, and the device does not know where to go to reach it, it will reach through the default gateway and hope for the best.
If there is no default gateway configured at all, the PC will only be able to reach local addresses, and addresses which have specific routes configured (where the path of routes never relies on a default gateway).
3] When an IP address changes by unplugging a router, will it auto change the ip addresses of every pc on the network?
Not by force. However, all computers using dynamic addresses that lose their connection (detect that the cable connection was lost, or lose wireless connection) will request a new lease when the router is available again. Some routers apply hashing and other techniques to assign the same address to one computer every time, others start from the first address, and hand out addresses in sequential order as requests come in.