\version "2.12.2"

fb = \figuremode {
  % Question 1: How can I add a figure above a rest??? It works above a skip!
  s4 <6 4> <5 3> <7 _!> |
  s4 <6 4> <5 3> <7 _!> |
  % Question 2: The figures collide with articulations (trill, staccato, accents)...
  <3> <3> <3> <3> |
  % Question 3: How can I draw a simple extender without a starting figure? (i.e. the default 5-3 chord stays)
  \bassFigureExtendersOn
  s4 <_> <_> <_> |
}

music = \relative c' {
  g4_"Figure above rest?" c r e |
  g,4 c s e |
  c-|_"Collisions" c\trill c-| c\accent |
  c,_"extender w/o starting figure" e g c, |
}

\new Score {
  \new Staff <<
    % NOTE: I can't use FiguredBass, because the figures should (1) be above
    %       the notes and (2) they should follow the skyline of the music
    %       instead of being all perfectly aligned vertically (but way too far
    %       away from the actual notes they belong to).
    \new Voice = "FiguredBass" { \fb }
    \new Voice = "Music" { \clef "bass" \music }
  >>
}