Assuming that there is no great WWIII or equivalent cataclysmic event the world of 2512 is beyond our faintest imaginings and would likely be frightening to us.
I don't speak about nanotechnology or even brain-uploading and synthetic sentience, I speak about rather more mundane trends that are almost certain to continue.
For example, manufacturing. Today manufacturing is still rather similar in nature to the way it was in the 17th century, we just have a whole crap-ton more of it and it's easier to ship manufactured goods around the globe. But I believe we are reaching an inflection point on manufacturing. We will soon reach a point where manufacturing becomes entirely automated for huge classes of devices. All you'll need to do is upload a set of files to a server somewhere and press a button and then a factory will produce whatever it is you've designed, on very short notice and in arbitrary volumes. This alone is a transformative technology, but let's take it a step further, toward fully automated creation of machine tools and to factories themselves. The idea of an assembly line as this huge, fixed entity is due to the nature of our manufacturing technology, but it's possible that manufacturing facilities will themselves become disposable (likely recyclable) and transient. Manufacturing won't be something that people consume, it will be something that people do. More so, the ability of a small amount of capital machinery to boot-strap into the manufacturing capabilities of a developed nation will rapidly eliminate almost all remaining undeveloped parts of the globe. Imagine what happens when you can ship a few containers of equipment to, say, antarctica and start building out factories, tractors, automobiles, houses, etc, etc. with only an input of crude raw materials.
How this will transform the world is beyond me, but it will certainly change our perception of wealth and scarcity and the people living in a world with this technology will be as unfamiliar to people of today as people of today would be to stone age tribes. And this technology is not a 500 year technology, it'll likely arrive in the next hundred years at most.
Let's talk about drugs and surgery and self. Modern medicine is at best a century old, and in some ways perhaps even less. There will come a time, certainly within the next 500 years, when medical technology in the realm of mood alteration, behavior alteration, and cosmetic surgery are at a level which we would describe from the perspective of today as nearly perfectly effective. Imagine what happens when people can change their personalities and their mental capabilities at whim? If you find you're depressed you can fix that, effectively and permanently. If you have a mental illness such as, say, schizophrenia or pedophilia then you can fix that too. And if you are dissatisfied with your mood or your personality you can change that too. Do you want to be an alpha personality? Do you want to be a thrill-seeker? Do you want to be bubbly and happy all the time? Easy peasy. Do you dislike the way your face or body looks or works? You can change that too. You can have a stunningly attractive and physically fit body with ease, and you can look like a movie star.
To say that this will change society is a gross understatement. In many ways I think this will be a bigger challenge to the world than any other technological or environmental challenge. To be honest I think it will be a larger challenge for our species than even trying to co-exist with thermonuclear weapons.
As for space, I think it will affect our future a great deal but perhaps not as much as these other things. One thing a lot of people get wrong about space is imagining that it's hard. It's not, we've just been doing it very, very badly. For the same exact amount of money the world has spent on space so far we could have easily built orbital cities and moon bases housing hundreds. Not with revolutionary technology, not with some alternate and hugely more cost effective programs, but merely with applying proven and existing systems and technologies in a sensible way instead of the haphazard way we have done so the last 4 decades or so. For example, for the same cost as the Shuttle program we could have continued launching Saturn Vs (at least 150 of them) which would have allowed us to easily put living quarters for hundreds of astronauts in Earth orbit and to build out moon bases (or Mars bases, frankly) quite easily. There are two other important factors people miss. First, once you have a substantial off-Earth industry then it's no longer reliant on the cost of launch from Earth's surface. You only have to launch the equipment for an automated space mining operation once, afterward you only need to keep it operational. The potential return in terms of mass launched from Earth vs. resources returned to Earth or to Earth orbit could be a great many orders of magnitude (millions or billions), much like it is for mining equipment here on Earth. Second, the world of the future will be unimaginably wealthier than we are. The parts of the world which are today developed will be even wealthier in the future, and much of the developing world will have developed within the next 100 and certainly 500 years. Even without factoring in technological and industrial advances which could make orbital launch cheaper (incidentally, things which are already running at a rampant pace of advancement even today) the simple factor of having a much, much larger total economy will mean that the amount of resources for space exploration will be larger than today by a factor of tens to hundreds. The idea that this doesn't translate into a substantial permanent off-Earth human population is, to me, patently ridiculous.
Overall, the idea of trying to predict the world of 500 to even the tiniest degree is probably a losing prospect, but it should be an interesting ride regardless.