I would say that your best bet is going to be to do this using Ethernet.
It will of course involve a little initial outlay (buying the network cards, assuming you don't already have them), but makes it easy to install Linux on the laptop using just two floppies and the network.
You'll also have the major advantage of being able to very easily and quickly transfer files between the laptop and desktop and share a printer.
Check the redhat hardware compatibility guide for a PCMCIA network card for the laptop and a PCI network card for the desktop.
Buy the cards and a suitable length of CROSSOVER Cat 5 cable. If you're only networking two PC's you don't need a hub or a switch.
Install the card in the RedHat 7.1 desktop. Boot up and Kudzu should automatically detect and install the drivers.
When asked, give your desktop PC the address 192.168.0.1 with subnet mask of 255.255.255.0. You don't need a default gateway or anything else. Set it to be enabled on bootup.
You said that most services are installed on the desktop but are not active. Try starting apache as root:
/etc/init.d/apache start
(I think... I don't have mine set up as the default RedHat install)
Hopefully that will have worked. Try with a browser going to:
http://localhost/
and
http://192.168.0.1/
If both work, excellent. If the first does and the second doesn't, you may need to allow it through your firewall. As root, type:
lokkit
Go through the menu and set it to allow http in.
Find your RedHat disks. Get 7.3 if you can, otherwise just use your 7.1 disks.
Now, find your htdocs directory. I can't remember where RedHat puts it - it might be /var/http or /usr/http
Create a new directory, rh71 (or 3 if you got that)
mount each disk in turn and:
cp -r /mnt/cdrom/* /var/http/htdocs/rh73
(substituting any bits that need changing). Allow the next disk(s) to overwrite any duplicate files they find.
Check that everyone has read permissions to rh73 and below. You may need to:
chmod -R a+r /var/http/htdocs/rh73/
now get two blank floppies.
Label the first RedHat 7.1(3?) Network Boot. Put it in the drive.
type:
dd if=/var/http/htdocs/rh73/images/bootnet.img of=/dev/fd0
Label the next disk RedHat 7.1(3?) Network Driver Disk. After the first is finished copying, put the second in the drive and type:
dd if=/var/http/htdocs/rh73/images/drvnet.img of=/dev/fd0
Now we're ready for the laptop. Insert the PCMCIA card. Connect your crossover cat 5 cable directly between the laptop network card and that of the desktop.
Insert the Network boot disk and switch on the laptop.
type "linux dd" at the installation menu. You'll need your Network Driver disk now.
When choosing an installation method, choose http.
Set up your network card with the address 192.168.0.2, subnet mask 255.255.255.0 and default gateway 192.168.0.1 (this last bit will help if your want to share your modem on the desktop PC later on).
Enter the server to install from as 192.168.0.1, and the installation path as /rh73.
The rest of the installation you should be familiar with, except that it will be done in text mode - don't worry, it's harmless enough!
Hope this helps...!
Ian