Regarding turbulence models, at the moment only algebraic turbulence models are available with the built-in solver, but you can use RANS turbulence models with the external solvers OpenFOAM and SU2. OpenFOAM is free and open source, however needs to be installed separately depending on your system. If you are using a Windows PC you can download OpenFOAM from here:
https://sourceforge.net/projects/openfoam/files/v2012/OpenCFD-OpenFOAM-v2012-DP-mingw-crosscompiled-WindowsInstaller.exe/downloadand for Linux or Mac
https://www.openfoam.com/download/current-releaseonce installed and recognized the "OpenFOAM" button should be active (not greyed out) and solver available to use.
> Do we solve full N-S Equations using some discretizing scheme?
Yes, the full NSE are solved using the finite element method with the built-in solver, and FVM with OpenFOAM or SU2.
> On another note, when we use the option “Surface Plot”, and select pressure as the variable, it displays pressure everywhere in the flow field. The usage of the word surface pressure is a bit confusing because we tend to associate it as the pressure over a solid (model) surface.
Yes, I can see the confusion. I think for 3D models it might correspond better (as it refers to plotting an expression on a surface), but I think in your case you mean that surface pressure would be analogous to a line plot in 2D? You can also do line plots with the "Point/Line Evaluation..." option in the "Post" menu.