Officially, from a video and cinema standpoint, "2k" is 1920x1080, while "4k" is 3840x2160 and "8k" is 7680x4320. For cinema, where a wider image is pretty common, "2k" is usually 2048x1080, 4k is 4096x2160, and 8k is 8192x4320. There are some other variations on that. There are some TV "standards" that support a 2x increase in those resolutions, but are still lumped into the 2k, 4k, and 8k monikers.
Generally speaking, though, that's what 2k and 4k are.
As far as "1k" goes, I don't think there really is an official standard for that. There is 720p, which is actually 1280x720, and that's about 1 megapixel.
To me, one of the biggest immediate benefits of 4k is the ability to downsample. You can downsample 4k to 2k, and the detail, sharpness, noise quality, all of it gets much better (your packing 8mp into 2mp). That's an immediate benefit for those who are producing 2k output...so 4k input can be useful even if you don't produce 4k output.
The other benefit is the cropping and recomposition. You can crop to anything from just slightly less than 4k, to as you said as small as 720. That can be a huge bonus for a number of things. You can recompose in post, since if you frame widely enough, your scene might fill the middle "2k" region, so you would have tons of space around that region to reframe. You can also use that extra space to apply things like post-process image stabilization with Premier, and it can do a really good job as well (hand-held panning or tracking can look like it was on a dolly or some other kind of rig.)
Yeah, Sony made some mistakes with the A7r and A7s. I'm hoping whatever they release early next year is a big improvement.
I agree that improving codecs would be huge for Canon. If I got deeper into video, I'd really want RAW output. I've been playing with magic lantern lately. It supposedly supports RAW video out...that's something I'm going to have to try.