You have to pay for vSphere with its various modules and extra features but not to use the vSphere Client to connect to a free ESXi.
I think where you may be getting the license message from is although ESXi is free, you still need to request a free license key from VMWare.
Login to your ESXi box with vSphere Client and go to Configuration -> Licensed Features -> Edit.
If you are set to evaluation mode, that is what you are getting the license warning from.
VMWare should have emailed you a license key when you signed up on their website to download ESXi. If not, you can go through the download steps again and the license key should be on one of the pages.
For me, if I go to https://www.vmware.com/products/esxi/ hit Download, login with my free VMWare account, then on the page with all of the download links, at the top of the list is my ESXi License.
The reason you are seeing the license message about vSphere is that in the Evaluation mode, some of the extra features that are only available with vSphere are enabled, once you enter a free ESXi license, those will be disabled and you won't get prompted anymore.
Also, you can use the vCenter Converter in the standalone mode (runs off of your workstation) for free with ESXi. This tool is immensely useful for moving VMs on and off of ESXi. http://www.vmware.com/products/converter/.
In my opinion, since you've already got the Windows server running (I'm assuming it is W2k3/8) then I would simply install RRAS on the Windows server and setup a VPN by allowing PPTP/GRE through the firewall by port forwarding/MIPing/whatever your router calls it.
Another solution would be to see if your router supports DD-WRT and setup a VPN using it.
Another would be to open up the ports necessary for RDP and the vsphere client using port forwarding as well, but this wouldn't be as secure as an actual VPN.
Another would be to install OpenVPN: http://openvpn.net/ as your VPN server and use that.
There are lots of choices, obviously...so you'll just have to find what works best for you.
Best Answer
In short: no. There is no way to get hardware DMA from a virtual machine in a way that would allow you to interact directly with that VM from an external machine without any software.
On the second machine, however, it would be possible to use a stripped-down thin client OS (like Windows FLP, Windows CE, LTSP etc) which would emulate a "thin client" and then connect to the VM over RDP. This also allows you more flexibility in the kind of presentation server software you wish to use (X, RDP, Citrix, VNC, Tarentella etc.)