problems with foreach loop and intersections

Amusingly enough, when I copied your code to save as a TeX document to play around with then I tried to save it as pgfplotsforeach.tex only to discover that I already have a file with that name as the answer to pgfmathsetmacro isn't setting the macro. I'm guessing that you have also seen that question because you have the edeftemp{...}temp in your foreach which looks very like it came from that answer.

The issue is actually the same as in that answer: since pgfplots saves up commands until the end of the axis environment, you have to think very carefully about what will be known at that time. Expanding macros is only part of it, you also have to think about coordinate and path names. Fortunately, in a foreach loop then there is an easy way to make things unique - simply append the loop counter to the name (with one caveat: if looping over decimals then one should use the count key on the loop to get an integer and append that since coordinate names with periods in them are not a good idea).

You also need to noexpand all the path construction commands inside your edef, not just the first.

So here's a working version of your code (I think!):

documentclass[border=5pt,tikz]{standalone}
%url{https://tex.stackexchange.com/q/586094/86}
usepackage{pgfplots}
usepackage{tikz}
usepackage{pgfplotstable}

pgfplotsset{compat=newest}

usetikzlibrary{calc}
usetikzlibrary{intersections}

begin{document}
    
    defheight{10cm}
    defwidth{10cm}
    defsc{0.8}
    
    defxmin{0}
    defxmax{5}
    defymin{0}
    defymax{40000}    
    defenlarge{0.05}
    
    defGda{12244}
    defGdb{4773.48}
    defGdc{1861}      % Gdb/Gda*Gdb   
    defGpa{15526.95}
    defGpb{6053.38}   % Gdb/Gda*Gpb
    defaa{1.87}
    defab{1.76}
    defac{1.66}       % ab/aa*ab
    defreqa{2.67}
    defreqb{2.97}
    defreqc{3.30}     % reqb/reqa*reqb
    
    begin{tikzpicture}
        
        begin{axis}[
            height=height, 
            width=width, 
            scale only axis=true, 
            scale=sc, 
            enlarge x limits=enlarge, 
            enlarge y limits=enlarge,
            samples=500,
            domain=xmin:xmax,
            restrict y to domain=ymin:ymax,
            xlabel=$mathrm{R}$,
            ylabel=$mathrm{E(R)}$,
            ]
            
            addplot[
            name path global=A,
            color=black,
            smooth,
            ]
            (x,{Gda*(1-exp(-aa*(x-reqa)))^2});
            
            addplot[
            color=black,
            smooth,
            ]
            (x,{Gpa+Gdb*(1-exp(-ab*(x-reqb)))^2});
            
            addplot[
            color=black,
            smooth,
            ]
            (x,{Gpa+Gpb+Gdc*(1-exp(-ac*(x-reqc)))^2});

        foreach [evaluate=i as n using 216.92*(i + 0.5) - 2.07*(i + 0.5)^2] i in {0,4,...,20} {%
            edeftemp{noexpand%
               path[name path global=B-i] (axis cs:xmin,n) -- (axis cs:xmax,n); % <- here is my problem
    noexpandpath[name intersections={of=A and B-i},];
    noexpandcoordinate (A1-i)  at (intersection-1);
    noexpandcoordinate (B1-i)  at (intersection-2);
    noexpanddraw (axis cs:xmin,n) -- (axis cs:xmax,n);
    noexpanddraw[red,line width=0.5pt] (A1-i) -- (B1-i);
  }temp
        }

        end{axis}  
    end{tikzpicture}
end{document}

I've zoomed in on the relevant part, and coloured the lines in red to make them obvious.