I tried to record a CD 3 times Saturday. 3 times I got errors such as:

cdrecord: Input/output error. write_g1: scsi sendcmd: no error
CDB: 2A 00 00 01 0D 5E 00 00 1F 00
status: 0x2 (CHECK CONDITION)
Sense Bytes: 71 00 03 00 00 00 00 0A 00 00 00 00 0C 00 00 00
Sense Key: 0x3 Medium Error, deferred error, Segment 0
Sense Code: 0x0C Qual 0x00 (write error) Fru 0x0
Sense flags: Blk 0 (not valid)
cmd finished after 0.005s timeout 40s
write track data: error after 97009664 bytes
cdrecord: A write error occured.
cdrecord: Please properly read the error message above.
cdrecord: Input/output error. close track/session: scsi sendcmd: no error
CDB: 5B 00 02 00 00 00 00 00 00 00
status: 0x2 (CHECK CONDITION)
Sense Bytes: 70 00 03 00 00 00 00 0A 00 00 00 00 72 01 00 00
Sense Key: 0x3 Medium Error, Segment 0
Sense Code: 0x72 Qual 0x01 (session fixation error writing lead-in) Fru 0x0
Sense flags: Blk 0 (not valid)
cmd finished after 13.397s timeout 480s
cmd finished after 13.397s timeout 480s
cdrecord: Cannot fixate disk.

The media are Verbatim CD-Rs from a box of 50, most of which I have already used with no problem.

The computer is an eMachines e725-4520 laptop; the CDROM is Optiarc DVD RW AD-7580S FX20 PQ: 0 ANSI: 5

The software is Cdrecord-ProDVD-ProBD-Clone 2.01.01a57 (i686-pc-linux-gnu) Copyright (C) 1995-2009 Jörg Schilling

kernel 3.4.4

I can read CDs.

How can I tell whether my drive or disks are faulty?

It's relatively simple:

1) Try other CDs (other brands, other batches of the same brand)
2) Try other drives (external drives, other computers' drives, swap the CD drive out for a spare if you have one)

You'll either work out which one is going wrong or get a situation (e.g. the burning only works if you connect the IDE cable to a specific motherboard port) which suggests that the problem is not as simple as "my CD burner's broken" or "my CDs are broken".

I suspect you know this already. The ability to burn is not the same parts used to read.

Disks don't tend to be faulty for the most part. The ability of a burner to use that brand is more an issue.
You say you have used most so we can assume that the disks ought to be OK unless they are real old or subjected to high heat or maybe ozone.

Time came for my monthly backup-to-CD, a brain-wave flowed over me: the switch would distinguish a bad drive from a bad disk. Even dummy write failed, consistent with a defective drive.

You could always try taking a can of compressed air to the lens on the CD drive. My girlfriend's brother got a free laptop because the owner thought it was hopelessly busted (wouldn't detect bootable CDs and wouldn't boot Windows). Took a can of compressed air to the CD drive and it then detected and booted CDs no problem... Windows wouldn't boot because of a virus and we just nuked the whole partition and then the laptop worked fine.