I still have a Gateway / DNS question to solve and I have no (more) ideas
how to solve it myself.
You can see the situation in the attached picture. The BM-Server can ping by
name and by IP address, but the other server (Server3) can't.
Server3 can only ping by IP-address. Also the workstation (laptop) can't
resolve DNS names (nslookup, ping). But through HTTP Proxy I can visit
on the test laptop by typing URL-Names.

Here are my thoughts: I thought that all request to a website (via browser,
ping etc.) go through my internal DHCP / DNS Server (Server3).
If SERVER3 can't resolve the address he'll ask the BM-Server. The
DNS-Resolver of BM-Server is set to the public DNS Server of my ISP.
Are these thoughts / configurations in my attached picture wrong?

Thanks for your suggestions...