>"1. Slow Auto-Focus when used in hand for certain tasks"
Most modern cameras will focus relatively quickly if there's enough light (which it seems you should have), so that doesn't narrow the choices down all that much. Or you could focus manually with your current camera (especially in conjunction with #4 below).
>"2. 1 out of 3 images appear blurry"
Blur due to camera shake can happen with any camera if the shutter speed is slow enough, so that doesn't really narrow things down much. Although, some sort of image stabilization system can help somewhat. But the real solution with any camera, including your current one, is to use a faster shutter speed (which you should be able to do given the lights you have).
Blur due to things being out of focus can also happen with any camera, and the solution would be to use manual focus (again, in conjunction with #4 below) especially since you're shooting a static (unmoving) subject (mannequin).
>"3. Snapping large resolution images is always processing slow before taking the next shot"
First of all, this could be due to a slow memory card, not the camera. So I would check that before continuing.
Having said that, some cameras are better at this than others. One measurement you can look at is the sustained burst rate: the more shots the camera can take before slowing down, the better; and the less it slows down, the better. But eventually, most cameras will slow down, especially when taking full resolution RAW pics. So another thing you can do, even with your current camera, is lower the resolution, especially since your intended output is for the web. (i.e. even a "Retina" display iPad is only 3MP, so there's no reason you need to shoot full resolution 10MP images on your current camera.) And since you're shooting in a controlled environment, you should be able to get the shot pretty close "in camera", so you may also be able to shoot jpeg (which has less processing overhead) instead of RAW.
>"4. The LCD screen is poor quality, though it looks sharp on-board but when transferring the images to our laptop they come out least expected"
Depends on what you mean by "least expected", but most cameras allow the user to "zoom in" both while viewing a previously taken image and while manually focusing.
Another option would be to use wi-fi to transfer the images as they're shot to a tablet/laptop. And some cameras have remote control apps that duplicate the LCD display on a tablet.
>"5. Battery life sucks (assuming this could be improved by purchasing a second)"
This is a common complaint. Personally, I always budget for a second battery when buying a camera. But a couple of other options: look for a camera with an optical viewfinder (so you don't have to use the LCD); use manual focus or at least use "single shot" auto-focus (as opposed to "continuous" mode), so the motor doesn't have to move the lens elements as much as possible.
Or if you don't mind being "tethered", you can look for a camera with a power cord adapter. But you may want to make sure you can still mount the camera on your tripod while it's being powered via the cord.