I've only rarely dropped into this thread, but on the two times I have, been pretty impressed with some of the photos. Looks like there's some knowledgeable people around here in this thread.
Am sure people have come into here asking for advice on what camera to buy, as I'm about to do, so apologies.
There was a DLSR thread about a while ago, which probably still has some good information.
Basically, you can ask a hundred photographers half these questions and get a hundred different answers, but I'll try and give you some of my experience.
1) Yes, getting rid of "shutter lag" was what led me into DSLR cameras in the first place. Press the shutter and get an image that instant, it's a basic and I'm amazed that point-and-shoot cameras still struggle to get this right. Any DSLR or compact system camera (AKA mirrorless) will have effectively no shutter lag.
Most will also get you 5 frames per second or better.
Video tends to be added as standard these days, whether you need it or not. The technological difference between a still camera and a video camera is pretty small these days, so it tends to get bundled in by manufacturers as it costs them very little and sells a lot more kit.
2) Most lenses are not fish-eyes. In fact a good fish-eye lens for a DSLR will set you back quite a lot of money. What can happen is that you get barrel distortion when you are very close to your subject. So if you shoot portraits with a wide-angle lens, you'll sometimes see a fish-eye like effect. Any decent standard telephoto zoom will avoid this. Most SLRs in your price range will come with a kit-lens that fits the bill.
3) OK, this one touches on one of the great debates in digital photography, RAW v JPEG. If you shoot RAW then you are going to process the image yourself in software to your own preferences. If you shoot JPEG (like most phones or p&s cameras) then you leave that processing up to the camera's internal computer. To get an effect like B&W, you can either shoot in RAW and get the effect later by putting it through lightroom, photoshop or similar software OR you can shoot in JPEG, set the picture mode to B&W and the camera will try and do it for you. I'd recommend at least trying to do it yourself. You can also usually select RAW+JPG in camera which means that the camera will have a go at processing the file for you, but the original will be there if you decide you want to try it yourself later on.
When I do street photos (see above) what I tend to do is set the camera in RAW+JPG, with the picture mode on B&W so that I get a preview of the image in black and white, but I'll go back and process the RAW file properly myself in Lightroom later to get it how I want.
If you want to see B&W in the viewfinder, you'll need a camera with an EVF (electronic viewfinder) rather than a traditional optical pentaprism. That limits you to either a Sony SLT like the A77ii or A68 or a mirrorless camera. The SLT is Sony's own compromise between a DSLR and a mirrorless camera. The tech is great, but Sony have been more interested in developing their E-Mount range of mirrorless cameras over the last three years. Given how Sony have been behaving over the SLT range, it's likely that the whole system, and the A-Mount lenses it supports, may not be around much longer, so I'd be wary of pointing you in that direction.
4) Is another reason to shoot in RAW mode, where you can effectively set the contrast levels wherever you like. You might get lucky with a built-in B&W jpg generator, but it's a roll of the dice.
5) That's the holy grail right there. Good news, this is the main area most manufacturers have been focussed on for the last 4-5 years and the tech is miles away from where it used to be. As a rule of thumb smaller sensor or more megapixels = more noise in lower light conditions, but newer cameras will often massively outperform older ones here. The low-light king is the full-frame Sony A7sII but that's a little over your budget.
6) Good to know. I think I can make a recommendation.
If I were in your shoes, I'd be looking at a mirrorless system camera. They are lightweight, but fully functional cameras that can do more or less anything a DSLR can do, except for professional sport photography. (The auto-focus tends to be slightly slower, but not enough for you to notice in ordinary use.)
You want something with a decent sensor size for that low-light stuff, which counts against M4/3 systems. (Although definitely have a look at them.) You want something with a viewfinder, which narrows that down further.
I'd be looking at the Sony E-Mount line, specifically the A6000. There's a new version (probably called A6100) coming out in the Spring, which will bring the price down a fair bit.
A great thing about E-Mount cameras is that you can use pretty much any lens you want with them. You can fit old manual lenses that you can pick up for very little on Ebay or at car-boot sales and get great quality optics at very low prices. If you don't mind mucking about with manual aperture settings (which are much simpler than they sound) then you have a world of great lenses available for next to nothing.
But even if you don't want to do that, the A6000 will come with a kit lens that should keep you going for quite a while by itself. It has an EVF so you can see what you're shooting in B&W if you like. It shoots in RAW+JPG and has all the features you'd want in a DSLR but in a smaller, more compact body. I'd suggest going on youtube and looking up Jason Lanier, who is a pro photographer who basically gave up his DSLR cameras to shoot with the A6000 and raves about it.
(If you're trying to share pictures here, it's usually best to upload it somewhere like photobucket and then post a link using "insert image" rather than upload the pic direct to RAWK itself.)