Linking Madhyamaka emptiness to Theravada emptiness through papanca

Short answer is you seem very close to what I think, however I might quibble with the way you use words. It is very hard to communicate precisely about the doctrine of emptiness so agreeing upon a definite jargon becomes ever more critical to picking out any actual differences vs just apparent differences. This is why the monks and nuns in Gelug monasteries spend years nailing down the jargon and conventions and mutually agreeing to it. Then they spend many more years debating each other in order to more firmly establish the conceptual understanding of emptiness.

"This is interesting, because it does not mean that a chair, a dog and Nirvana are mind-independently unreal or non-existent according to Madhyamaka. Rather, the mental idea that I have of a chair, a dog and Nirvana is unreal or non-existent."

Purported prasangika perspective... all errors or misunderstandings are my own.

This seems the most problematic part of your question to me. A chair, a dog and Nirvana are indeed existent according to Madhyamaka. However, from the ultimate perspective they are indeed unreal just like a dream or illusion is unreal. They have no true existence whatsoever. They only exist conventionally. It is not like, if you take away all the reification and elaboration that there is some real or truly existing thing that 'dog' or 'chair' or 'Nirvana' points to. Thinking otherwise is itself an example of reification or objectification.

This is very hard for us to understand as we hear the above and think, "My god, if it only exists conventionally without being based on any real thing, then it must not really exist at all!" and then fall into nihilism or annihilationism. Yet, this is how things exist. It is the only possible way for things to exist.

To back all this up have a look at Lama Tsongkhapa in his Great Treatise describing the object of negation:

Question: How does ignorance superimpose intrinsic nature?

Reply: In general, there appear in Candrakırti’s texts many usages of verbal conventions such as “nature” or “essence” with regard to objects that exist only conventionally. However, here in the case of reification by ignorance, there is, with regard to objects, be they persons or other phenomena, a conception that those phenomena have ontological status—a way of existing—in and of themselves, without being posited through the force of an awareness. The referent object that is thus apprehended by that ignorant conception, the independent ontological status of those phenomena, is identified as a hypothetical “self” or “intrinsic nature.” For, Aryadeva’s Four Hundred Stanzas says:

All of this is without its own power; Therefore there is no self.

Commenting on this, Candrakırti’s Commentary on the “Four Hundred Stanzas” says:

It is that which exists essentially, intrinsically, autonomously, and without depending on another....

Thus, he says that those are synonyms. “Without depending on another” does not mean not depending on causes and conditions. Instead, “other” refers to a subject, i.e., a conventional consciousness, and something is said not to depend on another due to not being posited through the force of that conventional consciousness.

Therefore, “autonomously” refers to the nature of an object that has its own unique ontological status or manner of being. It is just this that is called “essence” or “intrinsic nature.”

He goes on to describe the famous snake and rope allegory and says that all things exist just like the snake in that allegory. That there is nothing on the side of the object (a rope) at all that makes it a snake. In other words, take a snake and a rope... both of them equally lack any intrinsic 'snake' nature. To whatever extent you can find a 'snake' nature in a rope, to that exact same extent you can find a 'snake' nature in a snake -> that is not. at. all. To really put an exclamation point on it: there is nothing on the side of the object that makes a snake a snake as opposed to a rope.

So, linking the Mahayana Madhyamaka emptiness to the Theravada emptiness, I can say that all phenomena is empty of a mentally reified intrinsic essence, where this reification or objectification-classification is rooted in "I am the thinker".

I think it is important to talk more about how this rooting in "I am the thinker" takes place. Here is a theory, but mind you I can't support any of this with authoritative sources.

The thing we reify most is ourselves. For all of our lives in cyclic existence we've identified first and foremost with this "I am the thinker." And on that basis we've carved out a piece and objectified it and thought, "this "I" is separate from everything else in a real and intrinsic way." On the basis of this little island of "I" we then carve up the rest and objectify this and that. Put another way, the "I" is at the center of our conventions. We conceive of everything else in relation to this "I" and on that basis it is the root of our objectification.