Am using this in tandem with RM's Metal Over Flesh and MEC troopers mods, and regarding the flamethrower interaction with other skills:

Flamethrower + Deadeye = low-accuracy (normal) shot that deals only 1 damage (abnormal) to the intended target. The attack animates normally using heat beam mode, but suffice to say that the actual impact of the skill is...underwhelming.

All three modes doesn't seem to burn certain targets like Berserkers. From what I've seen, it burns ADVENT enemies just fine, but some alien enemies are more finicky to ignite.

Flamethrower + any reaction shot skill (Hunter's Protocol, Reactive Targeting Sensors) = unreliable activation. This one, however, will need some more testing from my end.

Also, is it possible to add a standard Overwatch skill to the flamethrowers?

Unfortunately most of those skill bugs can't really be fixed unless the weapon is drastically changed, i assume the reason for those problems is that the weapon doesn't have the "standard shot" ability, the weapon itself also deals no damage at all(this is probably why that one ability deals 1 damage), instead the abilities have their own damage values. So really the problem here is that there isn't a "flamethrower class" for SPARKs that is made with this weapon in mind. Which is my fault but i really don't have enough imagination to make a class mod, or the coding ability to make all the perks needed for one. As for aliens not igniting, yeah i dunno what's up with that, things that are resistant to fire would be immune but i don't think Berserkers are.

And i guess theoretically a special overwatch skill could be made for the flamethrower, but i did that way back for the Hunter Rifles from my Pathfinders mod in Vanilla Xcom2, and it was like opening a can of worms. It caused a lot of issues, similar to ones you are already experiencing. On the upside, i think the Flamethrowers are really powerful even if they don't benefit from a lot of "weapon perks"