networking - Application / Device setup for remote network connections

My question is related to what I asked about here: Why is my connection refused when port-forwarding?

So I have a device hooked up to my router and I would like to be able to connect to this device from anywhere where I have an internet connection.

The device is a DVR with security cameras. It is meant to have this functionality.

I already configured my router to allow for the necessary ports at the devices address to be forwarded (at least I think I did everything right - see my other question)

So then when it comes to the configuration of the application, I go to the network settings, and I can select STATIC or DHCP. When I use STATIC, I can select all the IP addresses and such myself, when I go DHCP I can only change the ports, the IP addresses will be chosen by the app.

However, I think there might be something wrong with my app, because the IP addresses don't make sense, at least not if I compare it to the things I see on the configuration page for my router...

It says IP Address: 192.168.001.100 however my router has assigned 192.168.001.253 for this device!

subnet mask 255.255.255.000 seems to be in accordance with what my router says.

But gateway: 192.168.001.001 I think gateway should be the same address as I have to type in to get to my router, shouldn't it? And for me that's ending in 254

Finally it says the same number for DNS 1 which I have a feeling is not correct.

On my router I can see unter Services > Broadband > Status > Internet Details an entry for Primary DNS and Secondary DNS both which are IP addresses, but not so much the standard kind of 192.168.1.1, instead they are somewhat similar to my public IP (and I have a feeling I should not publish them here).

Also I should note that I had to use a Lan-Cable splitter because I ran out of plugs/ports on my router. So the device I am trying to connect is sharing a LAN cable with another device. I am using this kind of splitter

So here are my questions:

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  Why is my application messing up it's IP and such, when on DHCP?

  
  


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  Should I set up everything in STATIC mode and enter the information I get from my router?

  
  


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  With this method, can I safely use the DNS that I see on my router's configuration page?

  
  


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  Also I take it I have to use the private IP inside the network, not my public IP, right?

  
  

Thanks for any help, I'm not really savvy with all this network stuff...

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EDIT: Starting a bounty

All I want to know is what numbers do I need to plug into the app, so that it'll allow me to connect remotely?

What numbers should I plug in for IP, Subnet Mask, Gateway, DNS1 and DNS2?