One router, 2 machines, to OS, 2 different ext. IP's?

I even don't dare to ask for a hint, but as I am looking for clues may someone can help me, though reading me bsd handbook. It is about one machine running as media play studio with a debian distro and one machine running bsd 10.2 connected to one router.
By booting first the debian machine it gets connected properly to the net, no changes at all in no instance, nor the network manager neither the resolf.conf. The odd normal provider is giving me via dhcp an address. Looking it up, ifconfig e.g. yep, there it is.
Booting up the bsd machine at the same time or afterwards, this very machine doesen't seem to connect to anything.
My humble question is, who is in command now? The router with it pre-configured installation or the machine thats booting first. Second question is, how to have through the same router two different external IP's?
I would like to connect on the bsd-machine to another server, doing so in the resolv.conf, on the very same router. If someone has an idea this would help me a lot, thanks in advance.

Why do you need to have "two external IP's"?

The router will have one external IP address and normally you would then use different ports for each machine. Please explain further.

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I would like to have a walkaround for the daily provider on the bsd-machine, both working as desktop, no server. The so called media-studio play station may connect via the provider as normally, the bsd workstation should connect to a hand picked server. The way you telling me, and as seeing it on both running, the same external IP, right. This also means to me, internal LAN via cable, something I'd like to change. And the easiest way to do so in bsd is the resolv.conf. Thats the reason why. Sure there is the method to change the socks5, and maybe a proxy etc. But to cut a long story short, I just want to jump on starting the bsd machine to a chosen server, thats all.

I still don't quite understand what you are trying to do.

I am not a bsd expert but resolv.conf in generic Unix only tells the OS which DNS servers to use for name resolution.

You want a particular workstation to send its internet traffic to a "hand picked server". Is that server on your LAN or elsewhere (on the internet)?
If it's on your LAN then you would configure the workstation to use that server as a proxy server.

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That server is on the WAN, the net. And as the network manager or the program resolvconf, together with dchclient and the networkmanager usually sets it to default by starting up, I would like to have a workaround. So I don't want to set up a proxy, nor to aim to one, it is about a walkaround e.g. to my actual external IP (lets say 187.15.217.29) and the second machine just connects to any other. It ain't about VPN or tunneling it.
Testing it, thanks to youtube, PuTTY or in firefox socks5, as hicksd8 said, depending on the port, thanks.