2. I was aware of the IP address method (shown as a choice in the JRemote "Choose connection method" screen in the "Add a new server" setup process), but I when I enter the IP address for my AirPort Express (as shown in the AirPort Utility) in the JRemote new server setup process and tap the "Add and connect" button, I get a "Could not connect" error message from JRemote. (BTW, I did *not* put anything in the "Authentication" fields, because I'm *not* using MC's "Authentication" feature.) Am I doning something wrong here? [emphasis added]
You are definitely doing something wrong if you're trying to use your router's IP address to connect; you want to use your server's IP address (see below).
You definitely don't need to be connected to the internet in order to use JRemote, I use it without internet access at work all the time. You do, however, need a device acting as a wi-fi router/DHCP server or it won't work. Generally there's no such thing as a "direct wifi connection," a router/DHCP server needs to be in the middle handing out IP addresses (although some computers and devices can act as routers/DHCP servers, cutting out the middle man).
In order to use JRemote without internet access, you need:
a) a working DHCP server somewhere, and
b) to specify the IP address of the media *server* you want to connect to (not your wifi router, unless your server is running on your router for some reason). Using an access key won't work (because the access key requires an online server lookup). So if your mac mini (which is your server) is at 192.168.1.6, you'd want to enter "192.168.1.6" in the IP dialog in JRemote, not the address of your airport.
If I had to guess what the issue is in your case, I would guess that either:
1) you're using the wrong IP address (remember you want your media server's IP, not your router's IP), or
2) your DSL uplink has a router in it which is performing DHCP duties in your house, and your Airport express is configured as a pure access point and not a router/DHCP server, in which case you'll need to figure out how to reconfigure your hardware.