I don't think you can buy a student version of LR 5 any more.
You can buy the non-student version from third-party vendors like Amazon and B&H, or you can buy Lightroom 6/Lightroom CC from Adobe. Unless your operating system won't run LR 6/Lightroom CC, there's really no reason to get Lightroom 5 at this time.
Your license allows you to install LR on two computers at one time, with the same license serial number.
If it's not possible to get LR 5 student anymore, does that mean a purchase like this:
will not work somewhere along the line?
If I purchase any kind of LR download, can I download the software again at a future data for a different operating system?
You will notice that that teacher/student edition is "temporarily out of stock", which I think is a little misleading. Purchasing the academic edition requires validation on your educational credentials before you can get a serial number. But Adobe doesn't sell Lightroom 5 at all anymore. So I don't believe you will be able to get that teacher/student discount.
If you purchase Lightroom as a download, there is always a resource where you can download the same version again in the future. The key element is that you must record your serial number where you can get to it. A serial number is not something that you just use once and throw it away. It is the thing that you "own", and will always be your key to installing and activating Lightroom.
Well, maybe I was wrong, I know that Adobe doesn't sell student edition of Lightroom any more, but it seems like Amazon does -- except that this item is listed as "out of stock", and I am skeptical it will ever change to "in stock", we shall see.
Yes, you can always download your software again from Adobe.
Indeed, the DVD-ROM version has been out of stock for a while and there's no indication it will be sold again; the download versions (on the same page) are - apparently - still available. I'm sceptical too, which is why I wanted to find out for certain whether I would be able to use it. If you're confident that Adobe wouldn't provide the necessary product code, then that makes my decision simpler.
Re the download, does it make any difference if you want to (subsequently) download the same version of Lightroom but for a different operating system?
I managed to 'chat' to an Adobe sales rep, who confirmed that:
After purchasing a download version, both Mac and PC versions can be subsequently downloaded.
A student discount is no longer available for a perpetual license; Adobe only sells LR6 (I'm assuming they wouldn't offer a product code if a version of the software was bought elsewhere).