Gene, big questions, I'll try to answer briefly.
1. Snapping is using features to locate to and from. They're the fundamental element of feature-based modeling, snap points are features of the drawing. Open a new drawing in TC, there's nothing in it and the line tool is active. You can click anywhere and place the end of the line where you click, or you can activate a grid in the drawing and either use the grid snap from the toolbar or the the right-click local snap, or simply press "g" key, TC's Single Entry Keyboard Equivalent (SEKE) while hovering the cursor near a grid intersection. The grid is the drawing's first feature. Now that you've located one end of a line, you can finish it either by clicking anywhere, or by using the grid snap again, measuring off your line by grid increments.
Now that you've a line in the drawing, you've added four more features that you can snap to besides the grid: Vertices (vertexes), which are the endpoints of the line, keyboard V, the middle of the line, keyboard M, divide points, dividing the line into whatever division is set, keyboard D, and nearest point of the line to the cursor, keyboard N. You can draw more objects that are exactly related to the line and drawing from those snap features. Any further objects that you draw add their own features and snap points. There are more snaps that relate to circles and arcs (centre, quadrant, tangent) and their 3D derivatives, as well as those particular to lines and polygons, and to their 3D derivatives. Snaps are what makes CAD precise. Objects relate to one another exactly numerically, rather than visually approximate. You don't have to enter each of the numbers into an entry box because the program derives the exact locations from the snap point.
2. Place cursor on object means put the cursor onto the object that carries the snap point you're locating to. If it's got more than one of the snap features, for example vertices on a box, put the cursor closest to the one that you mean to snap to. The object doesn't have to be selected first. It can be for another reason, but it doesn't have to be, the snap features exist whether the object is selected or not. In the example you've quoted, if you've selected something to move to that snap point, you'll have to click on the reference point to pick it up, in order to move it.
Related to that, the D SEKE picks up the object's reference point. The object reference point can be placed anywhere in the drawing, or snapped to any feature in the drawing, then picking it up the point picks up the object and moves it the same amount as you move the point, even if point and object aren't visible in the field of view at the same time. Almost like teleportation, really...
3. What's supposed to happen? Pick an object up by clicking on its reference point, then use a snap or SEKE (provided that the snap feature exists within the snap aperature, the "detection range" of the cursor), that'll put the reference point onto the feature you've pointed it to. The object will follow it's reference point, ie moving the same amount as the reference point.
4. H finishes a new line perpendicular (at right angle across) an existing one. Click to startdraw a new line near any other line, put the cursor near either end point of that line and press H, and the new line will complete as a "crossbar" to the line you snapped H to. Snap features exist whether the object is selected or not, whether or not an object is selected depends on what you're doing to it. In the example you've quoted, if you've selected something to move to a snap point, you'll have to click on the reference point to pick it up, in order to move it.
5. Snaps are part of most actions because they're a "point and shoot" shorthand for the numerical coordinates of the feature. They're self-contained references, they don't have "effects" of their own, they help you effect actions, so many actions will finish with a snap or SEKE because the snap refers "to there", shorthand for "to point x,y,z".
6. SingleSnap means that there are no more variables, it's a self-contained reference. If you press L, you then have to use another click or snap to relocate the origin. If you're drawing a line, pressing any of the "singles" locates an end immediately to a snap point, no more determination is required.
I've tried to explain clearly, but it's harder than I thought, and I've failed the "briefly" test miserably.....