I already legally installed your CD, but my laptop died. My new laptop does NOT have a CD drive. Can I use the same code to download the software from your web site?

Yes.  It can be installed 5 times.

Yes you can install on 5 computers.

How to install Turbo Tax if you don't have a CD drive

https://ttlc.intuit.com/community/installing/help/i-bought-the-cd-but-my-computer-doesn-t-have-a-cd-drive/01/26450

I have the exact same problem as the OP. But when I try to download TurboTax on my new laptop, it gives me an error and says that Turbotax has already been installed by another account, Not sure what I'm doing wrong...

If you installed from the CD it seems you can not download it from online now.  You can't do both CD and download.   See the new info on this FAQ

https://ttlc.intuit.com/community/installing/help/i-bought-the-cd-but-my-computer-doesn-t-have-a-cd-drive/00/26450

So you need to call in to get the download pushed to your account. How to contact Turbo Tax
https://ttlc.intuit.com/community/using-turbotax/help/how-do-i-contact-turbotax/00/26991

What is the Turbo Tax phone number
https://ttlc.intuit.com/community/using-turbotax/help/what-is-the-turbotax-phone-number/00/25632

Thanks! I'll do that and post an update here for others.

Contacting TurboTax didn't solve my problem. It just sent me to the voice menu, and they ended up sending me a link to "how to download", and back to my original problem.

But luckily I had an old laptop with a CD/DVD drive and a USB drive. So I ended up just copying my TurboTax DVD to a USB drive on that laptop, and then used the USB to install on my new laptop. So it all worked out.