Ok, first of all typically when you disconnect the swaybar for off-roading, you do not completely remove it, you just disconnect it from the axle. The easiest way to do this is to remove the swaybar end links (a bar with bushings on each end that bolt to the swaybar on one end and the axle on the other). You can take the stock links off to achieve this but it tends to be pretty time consuming. To remedy this a lot of companies make quick disconnect kits. The ones I use are made by JKS. This basically replace the stock sway bar end links. If you're getting a lift of more than an inch or two, you'll need to replace your end links with longer ones anyway, so a lot of lift kits come with swaybar disconnects (although some cheap 'economy' lifts will just come with longer non-disconnecting links).
The way the JKS 'quicker' disconnects work is that they have a top part that mounts to the swaybar which consists of the upper bushing and end link and lower bushing and a bottom part that consists of a post that mounts to the axle. The lower bushing slides over the post that is mounted to the axle and is secured by a click pin which is a metal pin that slides through a hole and goes all the way through the end link and the post and "clicks" in place. When you want to disconnect the swaybar for offroad use, you pull the pin out and pop the lower bushing off the post. The whole link then can pivot on the upper bushing which is still connected to the swaybar. You swing the link up and out of the way and slide the lower bushing over a 2nd post which is mounted to the frame, then secure it with the click pin and you're good to go.
I hope I did a good enough job of explaining this. If you want to see what they look like you can go to JKS' website or if you want I can take some pictures tomorrow.
And to answer your other question, if the swaybar is still connected the two wheels can still flex independently somewhat just not nearly as much. The idea is to try to keep both wheels on the ground as much as possible for maximum traction, so you want as much flex as you can get without damaging the suspension (there is such a thing as too much flex).