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Let's say I want to use an Emperor's Children Rapture Battalion and this is what I've taken:

Core: Kakophoni (Lucius/Lord and 3-6 units of Noise Marines)
Auxiliary: Heldrake Terror Pack (2-4 Heldrakes)

All good at this point, correct?

What if I also want to take 1 unit of Possessed (no formation consists of 1 unit of Possessed), am I allowed to take units that don't belong in any formation in the same detachment? Does this break anything or nullify any of the formation benefits?

yes you still have formations, but you would then be unbound for the rest, so your one unit of possessed would not have any bonuses from being bound. ie no objective secured etc.

Either your whole army is taken from a formation/detachment and therefore bound, or none of it is.
You can't have a formation and the few units you want tacked on the end as unbound.

This image is always a good one:

Yes you can. Unbound allows for formations. Just not detachments. So a kaophoni formations and a terror pack formations along with a unit of possessed is a legal unbound army. You would NOT get the command benefits of the rapture batallion however since that is a detachment and thus not legal in a unbound army.

Staying bound means ALL models need to be in a formation or detachment, so you would not be able to take a single unit of possessed alongside a rapture batallion. the closest you could do is take a CAD as well as the battallion and take a unit of possessed there. ( or in an allied detachment but that would mean making another faction your primary)
Alternatively you could take another formation of possessed with your battalion but it would be standalone. So no battalion command benefits for those possessed. (unless you go for the favoured one as auxiliary of course)
Crimson slaughter has some small formations that have possessed, but you can't make those into Emperor's children formations. They are stuck with being Crimson Slaughter.
The black legion also has one with 2 units and a prince, but again, forced into Black Legion. But that's just that formation. You're free to mix legions in this way, just not in the same detachment/formation.

Yeah unfortunately for you

If you want your list to be bound, and take possessed you will need to do a Chaos Warband (so they get the benefit) or do a separate Combined arms detachment.

My mates are having the issue, where they need to acquire Warpsmiths to be able to take their vehicles, in these Chaos detachments