xparse / wrapfigure: intextsep doesn't affect the output when set inside a NewDocumentEnvironment

Every time you put wrapfigure immediately after item (which is used by question and part), it gets pushed down one line. (Interestingly rlap{...} gets pushed up one line.) The only way to fix this is to manually add vskip-baselineskip inside wrapfigure. It has nothing to do with intextsep.

documentclass{exam}
usepackage{graphbox,wrapfig,duckuments}

usepackage{xparse}

NewDocumentEnvironment{mywrapfigure}{O{}mO{wrapoverhang}mO{1pt}}{%
    setlength{intextsep}{#5}%  <<< has no effect
    wrapfigure[#1]{#2}[#3]{#4}%
}{%
    endwrapfigure%
}

begin{document}
    begin{mywrapfigure}{r}{0.3linewidth}
            includegraphics[width=linewidth]{example-image-a}
        end{mywrapfigure}
        blindduck[1-2]
    bigskip
    begin{questions}
    begin{minipage}[t]{dimexpr linewidth}
           question%theeverypar
                begin{mywrapfigure}{r}{0.3linewidth}
                    vskip-baselineskip
                    includegraphics[width=linewidth]{example-image-b}
                end{mywrapfigure}
                blindduck[1-2]
        end{minipage}
        bigskip%hrule
        begin{parts}
            begin{minipage}[t]{linewidth}
                part[4]%theeverypar
                    begin{mywrapfigure}{r}{0.3linewidth}
                        vskip-baselineskip
                        includegraphics[width=linewidth]{example-image-c}
                    end{mywrapfigure}
                    blindduck[1-2]
            end{minipage}
        end{parts}
    end{questions}
end{document}

Update 1

You should always use WFclear if you run out of text before wrapfigure is done. Note the use of the optional argument [9] for the second image. Also note that linewidth keeps getting smaller, making the images smaller as well.

documentclass{exam}
usepackage{graphbox,wrapfig,duckuments,blindtext}

usepackage{xparse}

NewDocumentEnvironment{mywrapfigure}{O{}mO{wrapoverhang}mO{1pt}}{%
    setlength{intextsep}{#5}%  <<< has no effect
    wrapfigure[#1]{#2}[#3]{#4}%
}{%
    endwrapfigure%
}

begin{document}
    begin{mywrapfigure}{r}{0.3linewidth}
        includegraphics[width=linewidth]{example-image-a}
    end{mywrapfigure}
There once was a very smart but sadly blind duck. When it was still a small duckling it was renowned for
its good vision. But sadly as the duck grew older it caught a sickness which caused its eyesight to worsen.
It became so bad, that the duck couldn't read the notes it once took containing much of inline math. Only
displayed equations remained legible. That annoyed the smart duck, as it wasn't able to do its research any
longer. It called for its underduckling and said:  Go, find me the best eye ducktor there is. He shall heal
me from my disease!"
    %blindduck[1]par
    bigskip
    WFclear
    begin{questions}
        begin{minipage}[t]{dimexpr linewidth}
            question
            begin{mywrapfigure}[9]{r}{0.3linewidth}
                vskip-baselineskip
                includegraphics[width=linewidth]{example-image-b}
            end{mywrapfigure}
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis
facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent
imperdiet mi nec ante. Donec ullamcorper, felis non sodales commodo,
lectus velit ultrices augue, a dignissim nibh lectus placerat pede. 
Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing 
elit. Duis fringilla tristique neque. Sed interdum libero ut metus.
Pellentesque placerat. Nam rutrum augue a leo. Morbi sed elit sit
amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus tellus, aliquet aliquam,
luctus a, egestas a, turpis. Mauris lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis accumsan
            %blindtext[1]
        end{minipage}
        bigskip%hrule
        begin{parts}
            begin{minipage}[t]{linewidth}
                part[4]%theeverypar
                begin{mywrapfigure}{r}{0.3linewidth}
                    vskip-baselineskip
                    includegraphics[width=linewidth]{example-image-c}
                end{mywrapfigure}
There once was a very smart but sadly blind duck. When it was still a small duckling it was renowned for
its good vision. But sadly as the duck grew older it caught a sickness which caused its eyesight to worsen.
It became so bad, that the duck couldn't read the notes it once took containing much of inline math. Only
displayed equations remained legible. That annoyed the smart duck, as it wasn't able to do its research any
longer. It called for its underduckling and said:  Go, find me the best eye ducktor there is. He shall heal
me from my disease!"
WFclear
            end{minipage}
        end{parts}
    end{questions}
end{document}