Is it possible to connect a wireless router to another access point?
Your situation is basically the same as all home users, ie: the ISP/Telco has a complex network of ACLs etc. connected to a modem/router which the end user plugs into and is seamlessly integrated into the ISPs network.
Similarly your solution can be quite simple -- you just need to funnel your local network into the Buildings' network via some sort of modem analog, which in your case is an 802.11 connection.
You have three options:
Purchase a router/WAP with two wireless interfaces. For instance, Mikrotik's line of products such as the RB400 -- two 802.11a/b/g and three 10/100 ports. Connect one to the Buildings WiFi as a client, configure the second as a local WAP for your room.
Use two separate devices to do the above. Configure one device as a wireless client of the Buildings' network, and plug the second into the LAN ports of the first device. This second device would be your local WAP.
Use a desktop as you suggest with two wireless cards (re-creating the option 1 scenario).
I have never encountered issues with a reputable wireless card supporting connections from multiple clients assuming the card supports master mode ( that is, being the AP ) ... to be honest, I'm not entirely sure I understand what that question even means as 802.11 as part of the definition is for multiple clients to connect to the same AP ... just be aware that not all 802.11 cards support master mode and thus can not be an AP. Which is perhaps what your question really is??
Consider looking at cards built with Atheros 802.11 chips.
Option 1 will require a smallish router OS learning curve and have the fewest long term problems. Option 2 is the most "hackerish" with the least configuration issues. Option 3 will require a smallish "linux interface configuration/routing" learning curve and have the burden of cost/hardware storage.
hope this gives you some ideas.
-daniel
Have a look at m0n0wall features. There are quite a few BSD and Linux projects aimed at creating a router for your exact situation. Most standard hardware will be supported by most if not all of the solutions, each project will have it's own limitations and advantages.
I suggested m0n0wall because it's what I use, no problems in over five years of continuous operation.