Help connecting the dots (oddly wired 4-way)

This lighting circuit is for the flood lights on the back of my house. They’re WAY up high (3 stories) and getting into the fixtures is difficult/dangerous - so I haven’t done it.

Three switches in the house:

Box 1 (Basement - furthest from lights):
3 way switch.
14/3 entering the box
The Red is run to the common screw (black)
It appears that this Red is always hot. No matter the positions of the other switches.
(so the black/white are clearly travelers, not neutrals)

4-way switch
Two Big fat grey wires enter the box. I think they’re a 12 gauge, but the outer sheathing is grey (not yellow like I’m used to for 12-gauge). If I had to guess this is some kind of outdoor wire. I don’t see these wires in the other two boxes, so they must be running up to the flood lights.
Grey cable #1 black/white are both run to black screws.
Grey cable #2 black/white are both run to the gold screws.

Box 3 (Upstairs)
14/2 and 14/3 enter the box
3-way switch
14/2 (white) is pigtailed to 14/3(red)
14/2 (black) runs to the common screw
14/3 (black/white) run to silver screws
I do not think this 14/3 is same wire as the 14/3 in Box 1, because it is not always hot.

So my main question is:
Can I inject an Inovelli switch somewhere? Not sure I can.

Second question is – what the heck did they do here?
I gather they must run the line wire from the panel directly to a fixture (super long run to the roof line, and the basement 3-way would’ve been WAY closer). But okay, they did that… then what? They somehow joined the 4-way from the light fixtures - and maybe tied those into the three way wire in the light fixture boxes? Seems crazy to me. Open to fun speculation here. Thanks!

Any chance you can draw this up or post some pics to make it easier to follow? And if possible include what you’re seeing for wiring at the fixture itself, that will help.

The grey sheath on Romex indicates that it is UF (Underground Feeder), or like you thought, suitable for wet or underground locations. They are “fat” as you put it, because the sheath is solid around the conductors to provide the water resistance. Unlike the other colors, grey doesn’t indicate the gauge, so you can have #12 or #14 UF and they’ll both be grey.