I have two wireless devices. One device will connect to the Internet (wlan1). The other device will become the WAP (wlan0). I want computers that connect to the WAP (wlan0) to be able to access the server and share the wireless Internet access.

How do I accomplish this?

Basically, I'm try to obtain Internet Connection Sharing but with wireless devices.

In a sense, I'm turning my server into a wireless router, right?

With a simple device, such as a USB wi-fi stick, am I able to wirelessly connect more than one computer to wlan0 at a time?

Look into bridge mode. It may depend on the wireless drivers you have. The kernel supports bridging, but it must also be support by the driver. There might be other ways.

If this is going to be a permanent setup, it may be much simpler just to buy a wireless router that can be configure in bridge mode. That's what I do at home.