Ok, your networking issue is because you are running 2 dhcp servers. In order for lmce to get an internet IP on the external interface, you need that nic plugged into your existing router. The internal interface needs to be on a seperate switch. Otherwise it may have the opportunity to override the existing routers dhcp server. LMCE system was designed to replace or act as your home router to the internet. I had this issue initially. How I resolved it was disabling the dhcp server on my linksys router, plugging the internet modem directly into the core's external nic, and then plugging the core's internal nic into the router. This used the core as the dhcp server, and the router for the internet (the way it was meant to be used as) and because the internal nic is plugged into the linksys router, now all my wireless and wired connections can get to the internet via the core.
So you have 2 choices.
1. Get an addition switch for your internal lmce network, plug any MD's into this, and plug the core external interface into the router.
2. Plug your internet modem into the core external interface (eth0) and disable the dhcp server on your router, and plug the internal interface (eth1) into the router. Do not worry about firewalls/protection the core has all that included in its software platform.
As to your remote desktop reference, In order for this to work, you need to use option 2 for your network setup. This will give your laptop an ip from the core, allowing you to be able to communicate with it. You will need to install vnc on the core, and then after getting an ip assigned from the core on your laptop, use the windows software to connect to the core.
Hope this helps.