The Linux parts I can't help you with, but we do have other folks here who run Linux. I get beat up for using a Mac sometimes, so the "how to make it run" part you'll have to figure out.
There is no third party software that directly connects to the FLUX Delta, that's not quite how it works.
@goldensnake Luan is our resident guru with S3D and can help you out with the (un)Official profile. If I'm not mistaken he's been testing it for them and it is about ready for release with next version or something like that.
That gets you the machine profile in S3D. Then you go through the rest of the steps setting up your print inside S3D, importing the model, doing any rotating, scaling, adding support, etc.
S3D gives you a ton of options for setting up processes (and multiple process too).
The basic process, place on build plate, rotate, scale, add support. adjust settings for layers, temps, rafts cooling then Slice It. Save the file as gcode.
Then you open FLUX Studio and open the memory card on the top right and upload that gcode to the internal SD card. It will give you a message about converting the gcode to fcode. Click ok, wait for it to do it's thing, and then you'll see your converted file as fcode on the SD card. Double click it to start printing it with the settings you set up in S3D.
The real power comes in that you get a LOT more control. You can set different temps and different cooling for every single layer in a print if you wanted to. You can cut down on large print times by using multiple processes too, and use 30% infill at the base of a print, switch to 10% through the center and back to 30% again at the top, which can cut hours off.
For me personally, the biggest benefit is the ability to place and manipulate custom supports.
Definitely talk to our own @goldensnake here, and watch Maker's Muse (Angus Deveson) on YouTube, he's got a bunch of great videos on S3D.