Fan control & 3-way light switch combo

I have a wiring configuration that appears to allow for a single switch piece that controls a ceiling fan AND 3-way switching of a ceiling fan light (there’s another light switch on the other side of the room that seems to pair with the 3-way light switch).

What is currently hooked up in this spot is just a fan control (with a red and white wire), which controls the current lightless fan. However, there is another loose red wire in the box that goes hot with the light switch across the room, so I’m thinking I can replace the fan with one that does have a light that I can control from both locations.

My primary questions are:

1. Does it seem like I have correctly deduced that I can add a fan with a light in this way?
2. Is there a slider/switch combo I can purchase that both controls the fan and acts as a 3-way light control? I’m having trouble finding one, but might not be searching correctly

Bonus question: the fan is on a very high lofted ceiling – and some have said having a light there would get annoying when bulbs need to be changed (pretty much would need to rent a 12-ft ladder), however I think the extra lighting is worth it and bulbs these days last a long time… any thoughts or experiences? 😀

Thanks!!

UPDATE: adding photos of the two boxes. The larger box includes the current fan control hooked up on the very left, and the smaller box is the one wired for 3-way. The middle switch in the large box is a single-pole for a hall light and the one on the right is a 3-way switch paired with a different switch for another hall light. The two wires hooked up to the fan control (one red and one white) are both hot (per my contactless voltage tester): the white wire is coming from the cable on the far left (along with the loose red wire that goes hot when i flip the paired 3-way switch across the room, a ground wire, and a black wire that’s joined with the other black wires in the box); the red wire is coming from the second cable from the left in the box, along with a ground wire and white and black wires that are joined with their other like-colored wires in the box.

UPDATE2: adding a photo with more detail of the larger box and the wires labeled – source cables are colored in yellow and the corresponding individual wire terminuses in green.

UPDATE3: adding a chart of which wires are hot in the 3-way switch box and the larger box (just those on the left side: the red and white hooked up to the fan control and the other loose red wire), based on configuration of the 3-way switch I have installed (whether the red or the white wire is wired on the same side of the switch as the black common wire, and which position the switch is in). Checkmark=hot. Maybe this isn’t supposed to be a 3-way switch?