Run several prepared queries

The problem you suffer is because you are releasing the resources and memory associated with storing the result with $stmt->free_result() before making use of said data:

  

Note:

     

You should always release the result with mysqli_free_result (), when   the object of the result is no longer necessary.

Also, you should do the bind_result() after execute() and the store_result() before% % co:

  

Note:

     

Notice that all columns must be linked after   call mysqli_stmt_execute () and before calling   mysqli_stmt_fetch (). Depending on the column types, the variables   they can silently change to the corresponding PHP type.

The correct way to use bind_result() would be at the end of the work with the obtained records, which could be after obtaining all with free_result() or at the end of the fetch_all loop.

Here is an example of how to use the first query, use it and release it later:

 function datos(){

 $opcion=$this->opcion;
 $fila="";
 $dias="";
 if ($opcion=="todo") {
  echo "entre al if";
   $sql="SELECT count(e.RUT_PARTICIPANTE) as cantidad,cg.NOMBRE_CURSO,cg.COSTO,cu.ID_CURSO_UNICO,cu.DURACION_H,cu.JORNADA_H,
          cu.CANTIDAD_PARTICIPANTES,cu.fecha_inicio,cu.fecha_termino,cu.HORA_DESDE,cu.HORA_HASTA,c.nombre_comuna,cu.RUT_RELATOR,nombre_estado
          from  participanes p, enrolamiento e,curso_unico cu,curso_generico cg,comunas c,estado_curso ec
          where p.RUT_PARTICIPANTE=e.RUT_PARTICIPANTE
          AND   c.id_comuna=cu.id_comuna
          AND   cg.ID_CURSO=cu.ID_CURSO
          AND   cu.ID_CURSO_UNICO=e.ID_CURSO_UNICO
          AND ec.id_estado_curso=cu.id_estado_curso";
          $stmt=$this->conexion->prepare($sql);
          if (!$stmt) {
            echo $this->conexion->error;
          }
          $stmt->execute();
          $stmt->store_result(); 
          $stmt->bind_result($cantidad, $n_curso, $costo, $id_curso,
               $duracion, $jornada, $can_part, $f_inicio, $f_termino,
               $h_desde, $h_hasta, $comuna, $relator, $estado);



          while($stmt->fetch()){

            $stmt1=$this->conexion->prepare("SELECT d.nombre FROM curso_dias cd, dias d where cd.num_dia=d.num_dia and cd.ID_CURSO_UNICO=?");
            if (!$stmt1) {
              echo $this->conexion->error;
            }
            $stmt1->bind_param('i',$id_curso);
            $stmt1->execute();
            /* Una forma tosca de hacerlo ...
            $stmt1->store_result();
            $stmt1->bind_result($nombre);
            $dias = []; // o $dias = array(); si usas una versión antigua de PHP
            while ($stmt1->fetch()) {
                 $dias[] = $nombre;
            }
            $dias = implode(',', $dias); */
            /* La forma que te propongo: */
            $res = $stmt1->get_result();
            $dias = implode(',', $res->fetch_all());
            $stmt1->close();
            $horario=$dias."<br>desde:$h_desde hasta:$h_hasta";
           $fila.="<tr>
                   <td>$id_curso</td>
                   <td>$relator</td>
                   <td>$duracion</td>
                   <td>$jornada</td>
                   <td>$$cant_part</td>
                   <td>$f_inicio</td>
                   <td>$$f_termino</td>
                   <td>$horario</td>
                    <td>$estado</td>
                  </tr>";
            $dias="";
            echo $id_curso;   
          }
          $stmt->free_result();
          /* Esto es un alias del anterior: $stmt->close(); */

 }

 return $fila;
}

I, personally, prefer to get row by row or all results to a variable and work with that data, I do not like to let something modify variables and risk causing side effects.

Note: in the while interior you are doing while . Would not it be while ($stmt1->execute()) ? I've made some changes for you to review, maybe it's the best way to do what you try.