Love the question!
The weather aspect is probably the one I'm least sure off. Like Vincent stated, I'm pretty sure not much would change unless the buildings are fairly compact.
1) I'm taking Montreal as an example here (so it might not be the best example)::
Take your current city downtown areas, it may not look like it necessarily from the ground floor, but buildings really aren't that close together. In the few areas that they are, also like Vincent stated, wind accelerates because of the lack of space. I imagine in your future city, buildings WILL be close together so that's definitely something to take into account (for any balconies etc...)
2) If your buildings aren't all the same height (with 1km+ buildings I'm guess the variety in height may possibly be increased, with some buildings being 500m tall, others 1200 etc...) and even without the height variety, on a building that is over a km tall, all the ventilation and heating waste etc.. might not all be possible to dump outside the building from its top like current skyscrapers generally are. If you have 30% of your city that dumps out water vapor and heat at around 500m, and 30% of your city is over 600m etc... than I can see that creating different levels of micro climates inside the city itself. (Again assuming the city's building are tightly grouped), the lower ground having lower mercury index temperatures but less wind (resulting in perceived higher temperatures?), the middle having lower temperature and lower perceived temperature because of the wind chill, and the higher portion having higher humidity, temperature and wind factor.
3) Not in link with the weather directly. But don't forget the underground part of your city. Again I'll use Montreal as an example. The entire downtown is linked by underground complexes with multiple sub levels, being malls, transit lines, parking lots, food courts, utility services, etc... And those areas also kind of have their own 'weather'. In a futuristic city of your scope, I can imagine we'd be talking even more sub levels (5-10+?) which would impact the overall city even more.
EDIT: I did a bit of googling and thinking and buildings wouldn't just get taller, they'd also get bigger. So I think this would again probably lessen the climate changes (exterior weather). However with larger buildings of extreme heights, my guess is you'd end up with arcology type buildings, each one aiming to be as self-sufficient as possible. (Imagine you have 300 floors, how long it would take to get food at lunch, or the transit time it adds to go home etc...) Many Sci-Fi have such buildings - the most visual example I can think of (not the best but most visual) is the 'District' buildings in the Judge Dredd remake.
Which brings me to answer. The exterior weather would not be noticeably affected. However you'd have 'weather' inside your buildings - in direct reaction to the outside weather as well as anything going on inside. (Anyone who's worked in a large building knows how the heating/humidity/cooling systems can be fickle and temperamental - at least the one where I work is. The bigger the area to regulate, the more it'll be. Etc...)