For years cups on my four-hosts LAN has been very stable. Then I converted my Windows XP laptop to Puppy Linux. All went well until
I used the printer setup on Puppy. This overwrote the .conf files
on my Slackware cups server. Now, even with the puppy laptop removed
from ethernet my cups printing behaves as described in the subject
header. I deleted my printer (I have only one) and set everything up
again using only lpadmin (because I suspect the browser-based setup
on Puppy Linux caused the problem in the first place). and re-editing
printers.conf to what it was originally. - So now what to do?
Remove and re-install cups on the Slackware server? -At least I'll get
the latest version.

When jobs can't print they are queued until they can.

You should have a look in /var/spool/cups to see if anything is queued there.

Also if you have other systems send print jobs to this server then those other systems queue the jobs locally and only send one job at a time.

If you look at what is currently queued you can determine the source of the job and if it is an IP go look at the server with that IP to determine how many more jobs it has queued. At that point you can cancel on that other server to remove all the jobs it would otherwise send to your print server one at a time.

I have Slackware 10.2 with cups 1.1.23. Things were OK until
I put a Puppy Linux PC on my LAN. I've taken it away now but
still I get nothing when I attempt to print (eg via enscript asciifile).
The printer's LEDs show its online. Also I can command escputil
to get, say, ink levels.
My default (and only) printer is D92. This is the response I get from lpstat -t
------------------------
scheduler is running
system default destination: D92
device for D92: http://optiplex:631/printers/D92
D92 accepting requests since Jan 01 00:00
printer D92 now printing D92-10. enabled since Jan 01 00:00
Waiting for job to complete...
D92-10 root 13312 Wed 05 May 2010 01:22:24 AM BST
D92-11 remroot 13312 Wed 05 May 2010 01:22:24 AM BST
-----------------------
I get this sort of thing even after restarting cups (/etc/rc.d/rc.cups restart) and
even after I have removed all files (lots) from /var/spool/cups
Why remroot? I have no such user or group.

The "remroot" can come from the cups configuration files and can indicate that maybe another computer on your network is still trying to print some job.
I would try to delete all jobs on the server and on the cups clients ("cancel -a" as root) and see if the problem persists.

Did you look in /var/spool/cups? As I noted earlier the jobs queued there will include information as to where they came from.

If your job is D92-10 then in /var/spool/cups you ought to see a couple of files for it like:
c10
d10-001

If you run a "strings c10" you should see among other things the following information:

job-originating-user-name
<someusername>

job-originating-host-name
<host ip address>!

As I noted previously the submitting host is apt to have other jobs queued waiting for the one currently hung to finish before queuing them to your print server.