I use Trinity on Slackware as well as my Manjaro setup (I use that for games). In both cases I compile TDE from latest git sources whenever I think it might need a recompile due to system changes. Just did one in both distros a few days ago, in fact.

Trinity doesn't care about SystemD, that's back end gyrations. This is essentially 10 year old software that's still being maintained and it doesn't have that kind of integration or dependency. Note that distributors or distribution packagers might do different things for desktop integration if you install their packages. For example, I tried this on Kubuntu once from a PPA and found that the desktop integration was stepping on some of my global desktop config files and affecting Plasma 5 configuration... but that was there and then and there's none of that nonsense when I compile it)

As for Slackware, at this time you'd really want to download the latest slackware-current ISO and use that for your install. Slackware 14.2 is really too old in the guts (though it's still maintained with security patches, fixes and sometimes new minimum version upgrades to things so software like Firefox etc. can still build and run)