Does the writing speed when burning a CD or DVD affect the integrity of the data

Actually, yes. Faster a disk is burnt, higher is the chance the media may not have recorded it, and consequently higher is the chance of errors, especially with low/cheaper quality media. More detailed explanation here.

Probably preaching to the choir here, but this is experience gained after a couple decades of using the darn things. My first drive was a really expensive SCSI interface 1x CD-R. That $250 you pay for a really good manufacturer now? Pfft! I could buy six now for what I paid for that thing. Ouch!

1. Use a quality drive to produce the disk
2. Use a quality disk manufacturer (Mitsubishi, JVC, Sony) and as fresh a batch of disks as you can get.
3. Make sure the disk surface is clean before inserting into the drive
4. Slower burn speed is supposed to produce larger dots, verify data after burn
5. Multiple copies of important data is a "Really Good Idea™"

Also:

1. Time will degrade the write-ability of the dye layer, slow burn speeds will not make up for this, use as fresh a disk as you can get.
2. Heat will degrade the dye used in the writable layer.
3. Ultraviolet light will degrade the dye used in the writable layer.
4. Moisture can get through the label side coating and attack the aluminum flash layer.
5. In certain environments, there is a sort of mold that will eat the label side of the disk, even on pressed disks.

The 20-30 year lifespan quoted for writable disk life was based on advanced aging tests, not real lifetime tests. Real life shows that a high proportion of written disks probably aren't much good after 10 years.

So the name of the game is "make multiple copies over several media types for long term storage" on rather static data. With multiple copies, there's a chance of recovery of the bad files.

The case is not so good if large databases are involved, a couple bad spots can render the whole database useless on a couple disks. Most backups have a lifetime of weeks, so this isn't as much of an issue, but for monthly backups, you may need to do refresh copies every so often if long term storage is a legal liability.