The cost of improving the intersections of streets or avenues and spaces opposite alleys in an improvement district, except as specially provided in sections 16-609 to 16-624, shall be paid by the city as provided in sections 16-625 to 16-628. Nothing in sections 16-617 to 16-650 shall be construed to exempt any street or other railway company from improving, with such material as the mayor and city council may order, its whole right-of-way including all space between and one foot beyond the outer rails, at its own cost, whenever any street or avenue shall be ordered improved by the mayor and city council as provided by law. No street or other railway company shall enter upon or occupy any paved street or avenue, within five years after such paving shall have been completed, until it shall pay into the city treasury the original cost of paving between and one foot beyond the outer rails, which sum shall be credited on the special assessment upon the abutted lots. If the special assessment shall have been paid, then the money shall be paid, by warrant, to the party who has already paid such special assessment.