I'll answer part of my own question based on a new discovery. Part of my problem was that eth0 wasn't being brought up on boot (at all). In /etc/rc.d/rc?.d (where ? is 2, 5, and a couple others) there were these links:
i.e., the network was starting before the pcmcia card manager. I changed 'S45pcmcia' to 'S07pcmcia' so that the card manager would start before the network. Viola! I'm not sure if this is the correct fix, but it does seem to work, at least so far.
The other problem, not being able to use ppp when eth0 is up, is a bit more complicated. What I did discover was that (a) it's not a modem problem and (b) this problem doesn't exist if I assign eth0 a "local" IP address (e.g., 10.0.0.1) and use DNS server addresses that correspond to my local ISP. I'm not sure which of these actually was the problem.
This setup is ok since I don't plan to be on a "connected" network (Internet) and use the phone at the same time. I do plan to use the phone when I'm connected to two other pcs that I have at home, but those are not actually connected to the outside world.
So, it appears that the Xircom Cardbus Ethernet 10/100 + 56 works just fine with the Dell Inspiron 7500 and Mandrake Linux 7.0.