Whoooookay, that's a lot of questions! So:

If you abolish private property, then who has that property?

Everyone has it, referred to as "the commons"

Let’s imagine that it’s seized, by whom? How?

The State, which will still be necessary during this transitional period. It will use the power invested in it by the democratic process (because havenots vastly outnumber the haves) to take and redistribute the wealth. This will likely not be a peaceful process, but the creation of private property was not peaceful either!

And why wouldn’t that be thievery in the eyes of those who don’t want it? Because if I want it to happen, then it would be relinquishing, but if I don’t it would be coercive, because I cannot pay anything to that person, otherwise it would become a “haver” against all of those “havenotters” that gave their property for nothing but good will.

Property is theft.

The very existence of private land is based solely on violent theft from the commons. All land was stolen by armies and gifted to nobles and aristocrats and settlers and companies. The history of land acquisition is bloody all across the world.

Profit, too, is theft. Profit is the surplus value created by labor and unequal exchange through unfair trade and State-backed violence i.e. the market. Then those profits are used to buy politicians to steal even more, leading to endless wealth accumulation and evergrowing inequality.

And then there’s the redistribution fact, of how to do that? Equitable? By some principle? Depending on who you are and are not, you get X o Y amount of “property”?

Democratic decision making works here; we all own it together and we decide how it is used/distributed.

And then it’s the issue of how do you measure that “property”? Because two cups of sugar can be of similar value, but not two houses. It’s not the same to live in downtown Manhattan than in the middle of Saskatchewan.

Is it so unthinkable that we could democratically determine the value of a house based on location, quality, size, popularity, etc.? Why must this be done by a market? And before you say that the market is democratic, remember that democracy is "1 person, 1 vote" but the market gives the haves more votes than the havenots.

Finally, who does that? We? And who is “we”? Who organises “we”? How is “we” not anarchist? And if it’s anarchist, how do we ensure it’s just?

These are strategic questions. I don't think anarchism can achieve this because the State is necessary for the havenots to seize wealth and property and power from the haves. I worry that the State will need to be dealt with and won't just 'whither away' on its own, as posited by Marxists. Some kind of synthesis between the two? I don't know!

Ultimately, this is all just theory crafting divorced from real world struggle. Nothing you or I say on the internet really matters. We have to go out into the real world and test ideas against material reality, where things are ugly and messy

The world is already ugly and messy, though, so I think it's worth it.

We have nothing to lose but our chains.