Beware that many HG cassettes have the last 3-4 cogs grouped and held by a spider. No problem as you can file the whole spline on the spider, but there is a slight bugger that your spacing will be for that amount of gears. So If you get that 7speed cluster you may have to pick 6 of them fit and use your 13t with spacer built into to lock it. that will get you 7speed if the freehub is wide enough to actually reach the threads with your lockring cog.
If you have indexed shifter that may be a problem even if you put only 6cogs (5+ the last one), because you cannot use your spacers for the cluster of 3 cogs bolted together. But for those I hope you have friction shifters.
Another issue may be that your hub may be not long enough so using a 7speed cassette (with 6 cogs and the last one you have) may be too long to reach the threads for the locking, but you can always remove a cog of your choice to fit.

If you find cassettes where none of the cogs are bolted onto a spider is best.

In the parts bin at LBS I can find any cog I like, I just scrounge some of them that are in top condition with the desired tooth count and reuse my spacers. - that cost me next to nothing (a beer or two for the 7cogs I need for a cassette as i have 8speed). - a thing to look out is not getting cogs from 9-10 cassettes as they are thinner and I'd have to compensate with an extra spacer to maintain the overall spacing for 8speed (as i use indexed shifters)

As for how many splines are enough.. well one is enough but may feel sloppy and could damage the freehub. The rest that remain after you hack them is plenty to make any difference. (If you are keen you can file only part of that spline to match another spline like the others which are equidistant and same width all the way around)

As for the notation "t" with an arrow and index written - that's uniglide notation. Uniglide have twisted tooth instead of ramps like hg, so rotation does matter hence the arrow pointing the correct rotation, "index" was added there to know it works with indexed shifters (as it was a novelty item to have index shifters in the late 80's), and the number followed by T is the number of "T"ooth-count of that cog.