LineageOS for Samsung Galaxy Young (S6310)? I have a Samsung Galaxy Young (S6310), which I'm re-tasking to play with as a secondary, mini-tablet (with no cellular access i.e. no SIM card inside, just wifi for internet access). My issue is space available on the device for applications, data, etc.

I don't want to use an SD Card, because the one which was in this phone has already been moved to a new phone. Plus, I don't want to buy a new SD card, because the point of this exercise, for me, is to see if (without spending any money) I can take this already-four-year-old phone and get several more years out of it.

I've already removed old pictures/videos on it, old messages, rooted it to remove some unnecessary Samsung and Telstra bloatware, etc. But I'm still not having a whole lot of internal memory available, so my next step (unless I'm missing something which would make the next step unnecessary?) is flashing a different ROM onto the phone. I used CyanogenMod in the past on a different phone, so I'm thinking I might try out LineageOS, CyanogenMod's successor. However, herein lies where I got stuck.

Checking out the download page for LineageOS, after expanding the Samsung models on the left, I can't see the Galaxy Young or its model number in the list of devices at all.

Am I correct, and a ROM for the Samsung Galaxy Young is not available/supported by LineageOS? And if so, could I use a different ROM instead?

For the latter question, I'm assuming "no", but asking anyway in case the answer is actually more along the lines of "not really, but you could, with certain risks".
You didn't look hard enough, but it's probably not your fault, because 1) the site of LineageOS (LOS) does not list devices formerly in the CyanogenMod (CM) build roster, and 2) the device is never a member of the official CM/LOS queue.

There are a bunch of unofficial CM builds on XDA-Developers, all the way up to Android 5.x, with various bugs. See for yourself if you could accept them, and flash them with caution.

However, as for the problem you're attempting to solve (storage space), regrettably no custom ROMs will remedy that. Repartitioning will do, but 1) your device doesn't have a solution for such, and 2) being a low-end device, the storage chip itself is too small for repartitioning to claim space from.

TL;DR: "you could, with certain risks"