So do I understand correctly that a certain hox gene is activated in basically all cells which are in the "domain" of a certain vertebrae

Yes

and they all activate some subset of homeobox genes which in combination with the original hox gene cause them to start turning into all the different parts associated with that vertebrae (so organs and other structures)?

Not quite. The hox gene creates a protein that tells the nearby cells that they are in a specific segment. After this specific cells in that segment start signalling so they cooperatively lay out the cardinal directions to make that specific segment. In the shoulder segment, for example, a specific cell becomes the tip of the arm and tells all the cells about it with its signalling protein. All the cells in between it and the root now 'know' which part of the arm to grow.

This is a cascade of ever finer positioned 'location markers' that guide generic cells to specialise correctly.

Ultimately, as two bones grow into each other, they know to form a joint, and as that joint takes form the joint surfaces fit each other exactly.

Would we then need an entirely new hox gene to produce even a single gill? (I know you basically just laid out most of a response to this question.) Because I would assume although the exact point at which the development of our arms and legs begins is part of the whole hox gene "superstructure", but couldn't we 'basically just' highjack this same system and duplicate this gene to produce at least a single gill in the region where the current hox gene for our neck is expressed?

Assuming we want to keep our neck, jaw and ear features, we need to keep our existing hox gene and all the genes that turn on in this cascade to produce these structure. If we alter them, our development will change.

The issue is that in a fish or shark, exactly the same location marker is used to lay down their gills. So adding a shark hox gene will result in a human segment at that location. Hox is a marker - not the full set of instructions to build the segment.

We therefore need

1. A new location marker for the gill
2. And we need our developing cells to recognise this new signal
3. And we need a development pathway to create a gill which includes new location markers, and the ability for cells to differentiate in the right place to new tissues
4. New genes for specific proteins to create these new tissues (which may be copyable from other organisms)

Long story short: what is the biggest reason why we can't just hack into a later part of the sequence and continue on from there with what you said?

Well, we can't reuse the existing one because it creates human structure. So we need brand new genes for 2 and 3.

I'm not a professional in this area, but I haven't seen anything that suggests we can fo this yet.

I think part 4 (the bit about creating new tissues) might in fact be the easier part. But to cause them to be developed at the right time in the right place and at the correct size with brand new signals is waaaay out there.

Or would your proposed plan also just end up like this in the final product and you laid it out like this because it's already the most viable route into this mess? 😅

Speaking as someone whose last practical biology wiped out all the very expensive cell colonies, and that was 30 years ago, I hope my wild suggestions here are even vaguely in the right direction.