PN junction band gap - equal across all devices?
For a 100% pure element (no doping) it will be the same.

For a practical diode or transistor, the dopant levels and purity have quite a large effect on both the forward voltage and temperature coefficient.

For pure silicon the tempco is -2.1mV / degree C, but the venerable 2N3904 usually has -2mV / degree C (depends on manufacturer).

If you read the SPICE files for some devices, this can be inferred from ideality factor as seen in the Schockley diode equation. 

For a diode, this is N in the model.

For a transistor NF is the forward mode ideality factor and NR is the reverse ideality factor.

A perfect device will have N=1.
Here is the model for the MMBD4148 small signal / switching diode


  *SRC=MMBD4148;DI_MMBD4148;Diodes;Si;  75.0V  0.250A  4.00ns   Diodes Inc. Switching Diode 
  
  .MODEL DI_MMBD4148 D  ( IS=300n RS=0.422 BV=75.0 IBV=2.50u 
  + CJO=1.99p  M=0.333 N=2.77 TT=5.76n )


Here N is 2.77 (the higher this value, the less ideal it is)

Looking at the 2N3904 model


  
  
  
  Model Generated by MODPEX     *
  Copyright(c) Symmetry Design Systems
  All Rights Reserved        *
  UNPUBLISHED LICENSED SOFTWARE   *
  Contains Proprietary Information *
  Which is The Property of      *
  SYMMETRY OR ITS LICENSORS      *
  *Commercial Use or Resale Restricted *
  by Symmetry License Agreement    *
  
  
  Model generated on Aug 7, 01
  MODEL FORMAT: PSpice
  
  
  *$
  
  .MODEL Q2n3904 npn
  +IS=1.26532e-10 BF=206.302 NF=1.5 VAF=1000
  +IKF=0.0272221 ISE=2.30771e-09 NE=3.31052 BR=20.6302
  +NR=2.89609 VAR=9.39809 IKR=0.272221 ISC=2.30771e-09
  +NC=1.9876 RB=5.8376 IRB=50.3624 RBM=0.634251
  +RE=0.0001 RC=2.65711 XTB=0.1 XTI=1
  +EG=1.05 CJE=4.64214e-12 VJE=0.4 MJE=0.256227
  +TF=4.19578e-10 XTF=0.906167 VTF=8.75418 ITF=0.0105823
  +CJC=3.76961e-12 VJC=0.4 MJC=0.238109 XCJC=0.8
  +FC=0.512134 CJS=0 VJS=0.75 MJS=0.5
  +TR=6.82023e-08 PTF=0 KF=0 AF=1


Here the forward ideality factor is 1.5 (much closer to  an ideal diode). This is one of the reasons that temperature sensors often use a diode connected transistor.

So the forward voltage is process dependent (as is reverse leakage).