http://imgur.com/a/ivS9P
 
Regarding the gif above, oftentimes I am suddenly unable to split a clip, and I don't understand why. You can't see in the gif, but when I am showing the scissors cursor, I am repeatedly clicking on the clip to split it, but nothing happens. EDIT: After recording the gif, I learned that I CAN split the clip anywhere else, just not right where I want to. So I can split elsewhere and then move the ends of the split clips so they are split where I want them. But still, why?
 
Secondly, the gif shows me hovering over the end of the clip and only seeing that ubiquitous crossfade cursor with the two rectangles, which I NEVER need to use. Sometimes, and I don't understand why, it's the only editing mode available at the edge of a clip. I can't drag its end without also dragging other clips, and I also can't edit the fade envelope, even though there is clearly one there to edit.. Can someone help me get out of that common situation?

It appears that you are in take lanes. In this mode you are usually doing comping (selecting different takes from multiple take lanes to create one track). In this area in order to split clips you usually don't use the scissors but you just put your cursor towards the bottom of a clip and drag to make a new comp area. This will split all clips in all take lanes and unmute the one you dragged across. If you do want to split them you have to make sure that all the clips you want to split are selected first.
 
The two rectangles you see are to move a comping split point.  You can get to the slip edit for each clip but I usually have to make the take lanes larger and you put your cursor towards the top of the clip.  As for the fades, again because it appears you are in take lanes, you generally don't move them manually but you put your cursor between a split point and hold shift or control and the cursor will change allowing you to make the crossfade larger or smaller.
 
Please see the documentation: https://www.cakewalk.com/Documentation?product=SONAR&language=3&help=Arranging.42.html