Is the emptiness of final nirvana an illusion?

Perception isn't real, lasting, a refuge, proper to regard as me, myself and I. So, form, not to speak of any perception of form.

The Four Noble Truths are about suffering, not boxes.

So let's talk about loved ones. When a parent loses or misplaces a loved one, sorrow arises. Was the sorrow inside the loved one? The sorrow was not there before, so where did the sorrow come from? Trapped in suffering, what is the escape?

DN34:1.4.27: Renunciation is the escape from sensual pleasures. The formless is the escape from form. Cessation is the escape from whatever is created, conditioned, and dependently originated.

Looking for the end of suffering, we may become dissatisfied with form and eventually explore the formless, perceiving nothingness and emptiness:

MN121:8.7: There is only this that is not emptiness, namely the oneness dependent on the perception of the dimension of nothingness.’
MN121:8.8: And so they regard it as empty of what is not there, but as to what remains they understand that it is present.
MN121:8.9: That’s how emptiness is born in them—genuine, undistorted, and pure.

The truth to be found in boxes does not end sorrow, so we have to look beyond boxes to find the root of suffering. Going beyond boxes we might find a certain emptiness:

MN121:13.4: So, Ānanda, you should train like this: ‘We will enter and remain in the pure, ultimate, supreme emptiness.’
MN121:13.5: That’s how you should train.”
MN121:13.6: That is what the Buddha said.
MN121:13.7: Satisfied, Venerable Ānanda was happy with what the Buddha said.

nothingness is only really inside things that exist

Is surely the mistake.

Does everything exist that is composed of some but not all possible parts? Whether or not it is Buddhist, it is not trivially false, inconceivable, that the whole of reality exists, or that some parts do not combine.

It would be an unusual reading of interpenetration, and seems to amount to the claim that only and all absences exist. Which doesn't mean absences exist independent of anything, or that no theory is true of the possible whole - rather than actual, which does not exist.

Put another way (again): things that have arisen do not exist; things that cease do. An unusual way of looking at 'emptiness'

We have problems in our society, particularly when it comes to words, with the idea of "one". Concepts are bound up with this idea of duality as, for example, with your thoughts on existence and non-existenxe. The only alternative to this duality which we can conceive of seems to be nothing, nihilism. We bounce between "two" and "zero". Somehow "one" is a hot coal which we can only jump over without standing upon it. In some ways the idea of the self and the other are the root of this, the underlying duality, to the extent that has meaning. Ted Wrigley's excellent and creative answer, more worthy of votes than this one, illustrates this in a more useful way than the kind of conceptual thinking in this answer and in your question. But maybe this answer could serve as a bridge for you?

I'm not sure why you think of Nirvana as Nothingness.

I'm coming from a Theravada Background and Nothingness and Nirvana are not the same. There is no material plane, or place as Nirvana (Nibbana). The word Nibbana literally means severing the link or attachment (to sansara).

Nirvana is an understanding. That is why an Arahath is known as an 'Awakened one'. The understanding you attain in Arahath state allows you to stop forming attachments (lobha or dwesha) towards the world. You no longer have a reason to be born again, thus you have broken the cycle of birth and death. When an Arahath passes away there is no reason to form another link to the world, thus ending the birth and death cycle of sansara. It is not nothingness. If it helps you understand, you can think of it as ending of a process.

This perspective on emptiness is a bit turned around. Instead of standing as an outsider looking at a supposedly empty box, imagine being inside that box. It's (perhaps) a somewhat claustrophobic experience (though we might not recognize that feeling for what it is); we are in this small, confined space that is filled with 'us'. Then follow this progression:

• The box disappears and we find ourselves in a larger room, filled with 'our stuff'
• The walls of the room disappear, and we find ourselves in a larger house, intermingling with those we know and care about
• The house disappears and we find ourselves in a town or city, with all sorts of activity and distractions
• The town/city disappears and we find ourselves in an open field, with far horizons
• The field and sky disappear, and we are left in the vast expanses of space

We experience 'fullness' when we are bound by the attachments produced by the egoic mind, because we have placed ourselves in a mental container formed of our thoughts, attitudes, feelings, desires, etc. It's like wrapping ourselves in a blanket of mind; warm and familiar, but constrained. When we begin to quiet and still the egoic mind, those constraints loosen, and we start to feel the expanse around us. The more expansive that feeling becomes, the smaller and more intermittent the products of the egoic mind seem, and we start to experience that same sense of emptiness that we might have standing in a field under the wide blue sky. It isn't an objective emptiness in which nothing exists; it's the subjective emptiness of perspective and freedom.