Adjacent Additions

Answer: $25$


Using few unknowns
first + (middle 3) = (middle 3) + last $\therefore$ first = last, so the numbers indicated are equal
   
The code is:   

Repeat for the 5s:
   

The code is:    

Fours: $3a+x=16$
Fives: $4a+x=19$
$\therefore a=3$
All seven: $6a+x = 2a + (4a+x) = 6+19=25$


Using seven unknowns
Let the $7$-digit code be $abcdefg$.

We know that
$(1)$ $a+b+c+d=16$
$(2)$      $b+c+d+e=16$
$(3)$           $c+d+e+f=16$
$(4)$                $d+e+f+g=16$

$(5)$ $a+b+c+d+e=19$
$(6)$      $b+c+d+e+f=19$
$(7)$           $c+d+e+f+g=19$

If we take equation $(1)$ away from equation $(5)$ we obtain $e=3$.
Similarly:
$(6)-(2)$ gives that $f=3$,
$(7)-(3)$ gives that $g=3$,
$(5)-(2)$ gives that $a=3$,
$(6)-(3)$ gives that $b=3$ and
$(7)-(4)$ gives that $c=3$.

Then using equation $(1)$ we find that $d=7$.

So the code is $3337333$ and the sum of the digits is $25$.