|To connect a workstation to a Active Directory DC you need to have the workstation point to the AD DNS server.|
You do not use the ISP dns entries on the workstation in an AD environment.
You configure dns's forwarders tab to have the isp's dns servers. But there isn't even a connect to the server yet so dealing with forwarding or isp dns is very premature.
Static ip has NEVER been and NEVER will be a requirement on any lan nor does it address ANY issue except if you don't want to have a dhcp server. DHCP is just fine
dewald it does not appear you are following a MS training manual any more. The training would not direct you to use your router for dhcp. It would direct you to use the server for dhcp and disable the routers dhcp server.
you also should disable all other network interfaces, including hamachi except the wireless on both which is what you appear to be using until you have connected to the server.
Next step in troubleshooting would be to do a nslookup server01 from the workstation and post the results. This is a test to see if the workstation can talk to the servers dns and that the server dns will return the proper information.