Is it possible to create a functioning, prosperous economy without some kind of slavery?
Yes. 'Possible' means it could be done. Of course we can live without slavery and also be prosperous.
Has it been or can it ever be done?
No, but until Buddy Holly there was no such thing. We just need our Buddy Holly. You know what the UK has the distinction of being one of the first if not the first to outlaw owning humans. It may not be the first but is one of them. Put that in your museum.
Is there a prosperous nation that hasn't utilised some kind of slave system to make itself rich?
Yes: Liberia, though Liberia is currently plagued by dictatorships it certainly isn't prosperous. Technically yes. One could try to argue that Israel is prosperous without slaves, but I am not sure. I don't think its entirely slaveless, but I also don't think it has grown prosperous upon the backs of slaves. Like most countries it also has employees, so you can't say it isn't prospering from employees. It is.
Is it possible to build a nation from the ground without it?
Yes. Nations have a problem though. They are defensive devices. They are about keeping people out and in, controlling culture and controlling access to resources. The moment you make this country which has absolutely no employment indenture you make this country the most desired place to live. If no one has to work for anyone else it becomes the most valued place, and so you need either very effective conversion of immigrants or very stringent immigration controls or both. Western countries experience this to the degree that we are free. Western governments to varying degrees let people decide their own fates, so in some measure there is not required employment even though necessity forces employment upon most people. To the degree that there is freedom the western country will make immigration more difficult or enforce more stringent conversion or both, because nations protect resources, protect ownership.
I think a nation without ownership must control population having a very low population. People would need a lot more space so as to avoid conflicts over resources.
Is it within our human capacity to do so?
Within our capacity: yes. Likely? Not yet, but the idea is there. The idea is very powerful, too. It might help if we became less human and manipulated our genetics to have fur, stronger immune systems, more flexible joints and whatever would let us live without houses. As it is, we are quite vulnerable to the elements and uncomfortable in hot or cold weather; and that drives most of human misery.
Is it possible to live in the kind of materialist society we want to live in without some form of slavery?
Yes. The Sun provides enough energy to provide work for all automation needed, so there is a technological solution. Each human could have one or perhaps two or three non-sentient robot servants to do all of our work for us. Then we could find meaning through other means than work.