Can I use (share) a 120 leg of the 240 with the neutral to a 120 duplex receptacle in the same box coming off the 240 circuit breaker?
Check what breaker you have first!

Amazingly enough, this is actually permitted by Code under certain (common, but not universal) circumstances!

What you are creating is a multiwire branch circuit as per 210.4.  Normally, only line-to-neutral loads are allowed, but IF the breaker for this circuit is a two pole common trip unit (most two and three pole breakers are, although if the circuit is fed from the outer sections of a quadruplex breaker, you definitely can't do this as there is no way to get two common trips in that form factor), Exception 2 to 210.4(C) allows the circuit to feed both line-to-neutral and line-to-line loads (and even three phase loads for wye connected three phase MWBCs)


  210.4 Multiwire Branch Circuits.
  
  (A) General. Branch circuits recognized by this article shall
  be permitted as multiwire circuits. A multiwire circuit shall be
  permitted to be considered as multiple circuits. All conductors
  of a multiwire branch circuit shall originate from the same
  panelboard or similar distribution equipment.
  
  (B) Disconnecting Means. Each multiwire branch circuit
  shall be provided with a means that will simultaneously
  disconnect all ungrounded conductors at the point where
  the branch circuit originates.
  
  Informational Note: See 240.15(B) for information on the
  use of single-pole circuit breakers as the disconnecting means.
  
  (C) Line-to-Neutral Loads. Multiwire branch circuits
  shall supply only line-to-neutral loads.
  
  Exception No.1: A multiwire branch circuit that supplies
  only one utilization equipment.
  
  Exception No.2: Where all ungrounded conductors of the
  multiwire branch circuit are opened simultaneously by the
  branch-circuit overcurrent device.