Airfoil and Wing Analysis for Transonic Speed (M = 0.85 and Re = 42 million)
I am working on a project where I have to design a plane which has to
  cruise at 0.85 Mach at 35,000 ft. While working on the wing design, I
  quickly discovered that 6 series airfoils, with a good amount of
  thickness ratio, have a low Critical Mach Number. So, I started
  exploring Supercritical Airfoils (SC Airfoils). For doing analysis of
  SC airfoils can XFLR5 be used?


This is a direct copy-paste from XFOIL manual which is the underlying library for 2D analysis in XFLR5

One should always be wary of trusting solutions which show regions
of supersonic flow.  Such flows can be reliably predicted only with 
a truly nonlinear field method (such as the MSES code).  As a rule
of thumb, if the maximum Mach number doesn't exceed 1.05 anywhere,
shock losses will be very small, the Cp distributions will be 
reasonably accurate, and the drag predicted by XFOIL is likely
to be accurate.


  Moreover, Aspect Ratio and Sweep also influence Critical Mach Number.
  Is there are a relationship that connects these three entities?


AR does not influence the Critical Mach Number. However, sweep does
A direct copy-paste from old USENET message archive.

The airfoils on a swept wing behave as though they were flying at a
reduced speed, reduced Mach number, and reduced dynamic pressure.


effective speed = V cos(L)      
effective Mach  = M cos(L)     
effective q     = 0.5 rho V^2 [cos(L)]^2


where L is the sweep angle, and V and M are the airplane's speed and
Mach.


  Lastly: is there a credible source that compiles the analysis of
  Transonic Airfoils (Supercritical Airfoil)? (Similar to Theory of Wing
  Sections by Abbott and Doenhoff, which contains data for conventional
  airfoils)


NASA supercritical airfoils: A matrix of family-related airfoils


  I tried using XFLR5 for the analysis at Re = 42 million and 0.85 Mach
  Number. But, solution diverges


There was 2 issues I could see in the given dat file. 


finite TE gap and coarse coordinates.
I fixed this in XFOIL by going to GDES menu and setting the TGAP to 0 with the blending distance 1. 
And then used MDES to filter out glitches in the UE distribution. 
Mach effects.
Say your plane is flying at CL=0.6 but what happens when the MACH increases with constant alpha is, it improves CL. So the non-converging solution actually refers to a very high CL in the ranges of 3-4 which is not realistic and cp is high above the critical CP. Please refer to the images attached. when you directly use cl or reduce alpha accordingly you can get this sorted as shown on the last pic.