1. How can I move the (above and below) arrows a little closer to each other?, and
  2. How to put labels under the \faIcon symbols?

\node[scale=2.2]                        (A) at (0,0)        {\faUser};
\node[scale=2,right=0.75cm of A]        (B) at (A.east)     {\faInstitution};
\node[scale=2,right=0.75cm of B]        (C) at (B.east)     {\faInstitution};
\draw[gray,->] (A.north east)  --  (B.north west) node[midway, above]{\tiny{AA}}; 
\draw[gray,->] (B.north east)  --  (C.north west) node[midway, above]{\tiny {BB}}; ; 
\draw[gray,<-] (A.south east)  --  (B.south west) node[midway, below]{\tiny {CC}}; ; 
\draw[gray,<-] (B.south east)  --  (C.south west) node[midway, below]{\tiny {DD}}; ; 

enter image description here


With use of more libraries and style definitions the image code is much simpler:

\usetikzlibrary{arrows.meta,    % <--- changed
                chains,         % <--- new
                quotes}         % <--- new

node distance = 9mm,
  start chain = going right,
         N/.style = {inner ysep=0pt, font=\Huge, on chain},
        tc/.style = {transform canvas={yshift=#1}},
every label/.style= {font=\tiny, text=red},
every edge/.style = {draw, gray, -Straight Barb},
every edge quotes/.style = {auto, inner sep=1pt, font=\tiny}
\node[N,label=below:USER ] (A) {\faUser};
\node[N,label=below:ORG 1] (B) {\faInstitution};
\node[N,label=below:ORG 2] (C) {\faInstitution};
        (A)  edge ["AA"]    (B)
        (B)  edge ["BB"]    (C);
        (C)  edge ["AA"]    (B)
        (B)  edge ["BB"]    (A);

enter image description here


Please ensure that your example is minimal as well as working; your current example didn't compile for me. I tried to make a working example for your question. Feel free to let me know if you want something more.


\begin{tikzpicture}[label distance=-1em,every node/.style={font=\tiny}]
  (A) at (0,0)
  \node[scale=3,right=2em of A,label=below:{ORG1}]
  (B) at (A.east)
  \node[scale=3,right=2em of B,label=below:{ORG2}]
  (C) at (B.east)
  \draw[gray,->] (A.25)--(B.157)
  node[midway, above]{AA};
  \draw[gray,->] (B.25)--(C.155)
  node[midway, above]{BB};
  \draw[gray,<-] (A.335)--(B.203)
  node[midway, below]{CC};
  \draw[gray,<-] (B.335)--(C.205)
  node[midway, below]{DD};


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