Q1: That’s why it is called the Private Use Area, but some codepoints in the PUA are used by some commercial fonts.
These codepoints in the PUA area are used by FontCreator to generate Small Capitals, and selected ligatures.
57344-58031 Low Profile Accents
58033-58126 Basic Latin Small Capitals
58160-58255 Latin-1 Supplement Small Capitals
58256-58383 Latin Extended-A Small Capitals
58384-58591 Latin Extended-B Small Capitals
58912-58974 Basic Greek Small Capitals
59072-59218 Cyrillic Small Capitals
59680-59829 Latin Extended Additional Small Capitals
60531-60542 Number Forms, Nut Fractions or Stacking Fractions
61125-61148 Selected Ligatures from the Medieval Unicode Font Initiative
See the tutorials on Using Glyph Transformations and Complete Composites for details. You don’t need to worry about using these areas for your own purposes if you have no intentions of adding Small Caps, etc., to your font.
Q2: Insert Characters is the best way. There you can see which characters are already mapped, and which are available. You can also look for your planned symbol in the Unicode Character sets, such as Maths Symbol, Miscellaneous Technical, or Miscellaneous Symbol, to see if it already has an official codepoint. Using Insert Character adds the correct mappings for you.
Q3: The Font Test window uses a text file, which does not contain your new symbol. Select your new symbol in the Overview window and press “P” to see it in the preview toolbar. You can cut and paste it from there to the Font Test Window.
Q4: Could be several reasons for that. Format, Settings, Unicode Ranges, Calculate fixes most problems. Rebooting fixes some others.