1. The initial location of the marquee becomes 50% selected.
2. Outside the marquee the extent of selection fades away to zero as you get farther away from the initial marquee.
3. Inside the initial marquee the selection increases up to 100% selected as you move away from the marquee.
4. The final result marquee is placed at the zero percent selected point.
5. The width of the feather determines how fast the fading or increasing of the extent of selection occurs.

1. The original location of the marquee is unchanged in the result, always marking the zero-selected location.
2. There is a transition inside the marquee from zero selected to fully selected.
3. The width of this transition region depends on the Feather setting.

1. Everything inside the original marquee remains 100% selected.
2. A region outside the original marquee forms a transition from fully selected to completely unselected.
3. The width of this transition region depends on the Feather setting.
4. The result marquee is placed outside the original one at the
point where the extent of selection falls to zero.

I don't understand when changing from Dodge to Burn you get an error dialog that says you need to reset the prior tool before you can proceed?

I changed the setting to Always take these actions,
but I don't understand the need to have the dialog in the first place?

A number of the tools (such as retouch, Background Eraser, warp tools, etc.) use intermediate data that exist for the life of the tool. When you leave the tool and do something else (use another tool, run a filter) this transient data is discarded. The result is, when you return to the tool later, it might not give exactly the same results on the image as it did when you first ran it, since the memory of the original image is lost.  The warning is there to remind you of this.

We went several different ways on this, and decided that you would, by default, not get warnings for things you were unlikely to mind losing (e.g. Crop tool or Straighten tool settings), but would get warnings for things you might care about.

Some examples of why you might care are:

(1) if you jump in and out of the warp tools you will accumulate interpolation errors and things will start looking fuzzy and ugly;

(2) if you leave the Background Eraser and start unerasing with the regular Eraser you'll find bizarre colors that you don't get while unerasing with the Spacebar key depressed in the Background Eraser. (I've described this latter situation in detail in another post.)

Some of the consequences of losing transient data are rather subtle and you won't start stumbling on these gotchas until you have really dug into the program. In the meantime, these warnings must seem thoroughly puzzling and irrelevant.