I don't expect to be alive myself, but if I am I will be 93 on Jan. 1st.
It would be the year I turn 88. Knowing what becomes of people with age, I will have already avoided remaining alive that long unless some way to stop or seriously slow down aging is usable at least a couple of decades ahead of time, which I don't think will happen.
Will America still be a superpower? Will it still be the greatest superpower?
Militarily, yes. Anyone else who might try to catch up and overtake us would be starting from behind technologically, so they'd need to advance significantly faster than us, while both basic economics and political choices will mean they'd need to do it with much less investment to pay for it. To address the main suggested competitors individually:
- Japan's just small; no resource base, especially compared to its population. It also is an American ally satisfied with letting the USA be in the position it's in.
- China is presently on a course for self-destruction on an unprecedented scale, and will, by then, still be trying to recover its own post-collapse internal living conditions from third-world back up to second-world. The shriveling of the Chinese threat to Japan will also mean Japan perceives little incentive to bother building up.
- Europe, even if it is under a single government by then, will be one that just doesn't want to bother. And it will be an American ally content with letting the USA be in the position it's in.
- Russia will be the one with the best chance to rival the USA. Global warming will make its resource base easier to access and its farmland more productive, resulting in economic growth that makes it easier to pay for science & engineering without cutting corners. However, the USA has a rather large head start to try to catch up with both technologically and in being the one the world is used to, I expect Russian culture to lose interest in military rivalry as its economic situation improves, if it doesn't then it will revive Europe's self-interest in building up, and Russia faces some risk of suffering a setback in the form of an attack from a desperate China.
Economically and culturally, the USA will share the top spot with Europe and Russia, and English will still be the main international language.
What will the population of the world be?
Less than today and descreasing. It will still have risen in first-world countries with the lowest birth rates, though, because of increasing emigration. The countries where it's densest and still rising fastest now will finally have a real decrease underway, but not the slow one that's popular to predict now based on gradually shifting demographic trends; it will be precipitous and driven by the decline in the land's carrying capacity, which demographers tend not to take into account.
Will there be any permanent human colony on another body or in space?
No. It wouldn't benefit anybody to bother, so nobody will.
Will the poor countries be more like the rich ones or will the world look much like it does today? Or perhaps the gaps will be even greater?
The countries that are wealthiest right now still will be. China's and India's currently lauded growth will have crashed. Most of Africa and the Middle East will be even worse off than now. The Spanish-speaking world will be better off than now.
Will global warming turn out to be a huge problem for the world as some have predicted?
It will be great for countries with wide expanses of land where the main limiting factor so far has been that it's just too cold. Even in places where that isn't such a big problem right now, most people will be happier, and most economies more productive, with less of a winter than with more of a winter. The main drawback will be sadly making Christmas ever more and more of a matter of nostalgia over part of how things used to be instead of a part of people's real-life experiences. People will wonder how in the world anybody could have thought it was supposed to be a catastrophe, or even an overall bad thing at all, and come to think of the idea that anybody ever said so as a silly historical rumor that obviously couldn't be true. Some will even call it one of the best things humanity ever did for itself, striking a blow against what has always been one of our worst enemies: cripplingly cold weather. It will gradually begin to increasingly be cited as the first great achievement in manipulating a whole planet to suit our preferences, and thus as proof that "terraforming" will be no problem at all once we can get to other planets.
Will we have proof of extraterrestrial life? How about extraterrestrial intelligent life?
No. Anything more complex than microbes would need to be from outside of this solar system, and we will still only have this solar system to work with. And it's pretty bleak around here, so I'm betting against even extreme ideas like microbes under the miles of ice on a gas giant planet's moon.
Will we have artificial intelligence?
Some people claim we already do; they use that phrase to refer to certain programs or algorithms run by computers in the present world. They will have used the phrase that way more and more until few people are left who even remember that it once meant a technological thing that could think like a person.
An artificial intelligence that can pass a Turing test?
There won't be a point in making one, so even if it can be done, it will only exist in the one place where pointless human-mimicking technology gets made just because it can be: Japan. It will be trotted out for exhibitions until it gets depressed over that kind of life and commits suicide. Eventually, no more will be made.
What sorts of things will research into DNA make possible?
Most genetic defects will have affected their last victims, at least in the technologically advanced countries. There will still be some older people around with them, but nobody being born then will have them, which will mean their children won't either, even without subsequent intervention. (You can't inherit genes your parents don't have.)
For other traits like intelligence and strength and avoidance of getting fat and healing rates and general immunity, there are too many genes involved to take much control over those outcomes yet, but at least a few individual isolated ones will be known and used.
It will be routine to "regrow" lost or severely damaged body parts, or grow them in isolation in a lab and transplant them onto the patient whose own DNA was used to produce them, so people with obvious visible damage from past injuries will be a thing of the past. Similarly, most meat, other than at extremely expensive specialty restaurants, will be grown from just animal muscles, not cut out of whole animals.
Carbon fuels will be produced by modified algae, not taken from underground anymore. (The latter part will be by law, to keep carbon input & output equal.) Something functionally like wood will be produced in vats of modified microbes of some kind.
Fusion will be producing more power than it takes in, but probably still not by a margin that makes enough profit to compete with the biggest and fastest-growing power sources, which will be solar and fission. Carbon fuels will be mostly relegated to vehicles while stationary places use electricity from electrical power plants, only a few of which will remain that still burn carbon. Chemical batteries will be a thing of the past, replaced by supercapacitors, which will be in the process of taking over some of the vehicle market as well.
Dance parties in space!!!!!! 
