I have a CD that has lots of scratches, and I can see black spots all around the CD. I'm trying to rip it with EAC but it's really slow... Secure mode says it will take 6 hours for just one track. I tried ripping with burst mode but it gives skips.

I then polished the CD with toothpaste, and now the ripped tracks (with burst mode) don't skip but I get a different CRC every time! Comparing the wavs gives different samples throughtout the whole track for all the files.

If I just let secure mode run until it finishes, will it damage my drive considering the huge amount of time and error correction?

Short answer: YES, it will wear down your drive.

EAC's secure mode is based on a "read-go back-read" method that will put an extra stress on the drive.
Don't worry though, it would take much more than just a single 6-hour-ripping session to kill a modern drive.

If you do not this often, you will not notice the impact of this event on your drive's life.

I always turn on the automatic cool down feature, and then when discs like this are going to be ripped then it will shut down the drive for a while to let it cool, and then start back up later on. I've changed mine to cool for 7 minutes after every 30 minutes of ripping. I believe the defaults are 15 after every 60? Not sure that mine's actually any better, just seemed safer to me. Might not even matter.

As it has been said, it can damage your drive. My brother killed his one shortly before he finished ripping as many CDs to MP3 as it takes to fill the original Creative DAP Jukebox (6 GB) in secure mode, including scratched CDs. It developed read errors, he couldn't install games etc. anymore. The drive hasn't been used for much else than ripping...

Does setting the cooling period help prolong the lifespan?

It is believed that "cooling" (I would rather call it drive stand-by) helps on that issue ... but you may consider that there are already high temperatures in most modern pc casings ... so cooling down will only show results if the heat produced by the drive's motor can be transported out of the casing and this will only happen quickly enough if a significant temperature gradient is present from drive surface to pc internals ...

You might consider using a dvd drive for ripping of damaged CD's ... DVD drives are believed to be more suitable for long-term runs (over some 2 hours during DVD playback as an example) than CD-ROM drives ...

Do EAC report read errors ? If so, Try to rip a short part and see if anything is improved over burst mode. If not, it's not worth doing it. Secure mode was not designed to recover damaged CDs, just to warn that they are.

I'm not sure if EAC will give a read error (probably will), as I just cancelled it after 5 minutes of ripping seeing that it will take so long. But if I do 2 secure mode rips of the first few seconds of the first track, and compare them I get no differences found. Whereas if I use burst mode, differences are found in almost all the samples...

Try repolishing the read surface of the cd. There are numerous threads on ways to do this.