The answers per criterion:

1. In LR go through the photos in the Library module with the Loupe view using the arrow keys. Hit X when you want to discard a photo.

2. To keep the discarded photos from the list click the left and middle flag in the filter bar. I've added a screenshot showing the location of the buttons:
   
   This hides the discarded photos from the list and shows the unflagged and flagged photos.

3. If you pressed X accidentally you can hit cmd+Z to undo setting the reject flag.

4. Once you're done, go to All Photographs, press cmd+Backspace and LR will attempt to delete all discarded photographs marked with a reject flag. It will show you the following dialog:
   
   Press Delete from Disk and the discarded photos will be placed in the Finder Trash. Remember to empty the Trash to reclaim disk space.

Some additional remarks:

• Click on Photo -> Auto Advance to make the arrow keys obsolete. When you use the U, P and X keys to unflag, flag and reject a photo the selection advances automatically with this function enabled. So you don't have to use the arrow keys anymore.
• You can select multiple photos from the Grid or Loupe view and hit X to reject them simultaneously.

• When you want to have the filter applied at start-up do the following:

  1. In the Library module go to View -> Show Filter Bar. The filter bar will appear above your grid view:

  2. Click on Attribute and apply the same filters you did before (flagged + unflagged).

  3. Now click the little lock in the right corner of the filter bar, indicated with an arrow in the previous screenshot.