Hartree Fock and DFT - Exchange term
Most of the original question has been solved by discussion in the comments, but a pertinent question remains:


  [W]hy you do not simply take the HF expression for the exchange part and insert it into the DFT equations [?]


The answer is that it has been tried and does not work well. In a sense, the DFT quantities "exchange energy" ($x$) and "correlation energy" ($c$) do not quite match the HF quantities "exchange energy" and "correlation energy".$^1$ In fact, even within DFT, the quantities do not match between functionals, as can be seen in some $x$- and $c$-functionals working together well and some not so well. In particular, those that have been fitted in combination with each other often only work in that combination. This suggests to me an illusion of a clear-cut way to differentiate the two in the DFT framework. It is also true that in some functionals, the $x$-term appears to account for some static correlation (which, broadly speaking, often becomes important when a single Lewis structure does not describe the molecule well: transition states, resonance structures). Entirely replacing it with Fock exchange forgoes this advantage.

Some programs, such as NWChem, and to some extent Gaussian, allow you to combine functionals and Fock exchange (often with arbitrary weights) to conduct your own experiment. A small benchmark set for main-group thermochemistry would probably give you an indication.

I wish I could you references for this, but this has been related to me verbally by my advisor back in the day and I no longer have access to most of the literature. If anybody has such references, I would be happy to see them included.



$^1$ The notation is a bit sloppy, but we're not striving for rigor in this answer.