This is because 5 is an integer (int), and you're telling sprintf to pretend that it's a double-precision floating-point number (double). You need to change this:

sprintf(word,"%.9g", 5);

to either of these:

sprintf(word,"%.9g", 5.0);
sprintf(word,"%.9g", (double) 5);

use like this

sprintf(outputstr, "%+7.2f", double_number);

E.g.

#include <stdio.h>
#include <string.h>
#include <math.h>

void output_string(char output_buffer[], double nums[], size_t size){
/*  use '+' flag version
    int i,len=0;
    for(i=0;i<size;++i)
        len += sprintf(output_buffer + len, "%+7.2f\n", nums[i]);
*/  //handmade version
    int i, len=0;
    for(i=0;i<size;++i){
        char sign = nums[i] < 0 ? '-' : '+';
        char *signp;
        double temp = abs(nums[i]);

        len += sprintf(signp = output_buffer + len, "%7.2f\n", temp);
        signp[strcspn(signp, "0123456789")-1] = sign;//The width including the sign is secured
    }
}

int main(){
    double nums[] = {
        +300.00,
        -200.00,
         +34.60,
        +300.00,
        +233.45,
         -20.34
    };
    char output_buffer[1024];
    int size = sizeof(nums)/sizeof(*nums);

    output_string(output_buffer, nums, size);
    printf("%s", output_buffer);

    return 0;
}