Can I download lightroom classic on multiple devices?

Close, but not quite. You are correct when saying that what’s in the clouds is not the full file, but all of the edits ARE there. However, you can download the full file.

Nonetheless, let’s say that at home, you get a copy of LRClassic, or LR Desktop; one for her and one for you. You still could not run them at home at the same time.

Sorry, I wish there were a way, but you’d both need separate licenses. HOWEVER, you can get hers at a Student Discount. A bit of consulation. 

You can download LrC as many times as you want, now and in the future. You can have only two installations active at any one time.

However, LrC license is a single-user license. You cannot share your LrC license with others, even family members. Since you have now admitted here in a public forum that you are going to be sharing your LrC license with someone else, and presumably Adobe can monitor your number of installations, I would strongly recommend you pay for two licenses.

You can definitely install Lightroom Classic on more than two computers! You just cannot be logged in on more than two computers at the same time. If you try to login on the third, you’ll get a dialog asking you to logout of one of the other two (which you can do in that dialog), so this should go rather smoothly.

Because Lightroom Classic does indeed work with local images and a local catalog, the only way to use the same catalog on two computers is to have the catalog and the images on an external drive however, that you connect to the computer you want to work on. That may be a problem with a school computer…

Sorry, but two means two.

However, I am confused about “your daughter’s school laptop.” Is that provided by the school or is it your daughter’s? If the laptop is provided by the school, shouldn’t the school be providing the software?

Also, be aware that with Lightroom Classic, it makes no sense to have it on a desktop AND a laptop for one user unless ALL the images, catalogs and other related files are on a separate hard drive and are moved back and forth for the user. 

Lightroom Desktop is better if you want it on a Desktop and laptop computer because all of the files are in the clouds (It can also access files on the computer, but if they are left there, they will not be avialable on the other computer).

Why are you focusing in on LRC? Are there features you want in that that are not avialable on LRD (there are and vise versa)?