Any city of the third class shall have full power and authority, under the following conditions, to do the following things: To levy and collect taxes, for general revenue purposes, on all property within the limits of such city, taxable according to the laws of this state. To grade, pave (the word “pave” as herein used meaning to improve with all kinds of street paving, including macadamizing), gutter, curb and otherwise improve streets and alleys, and parts of same, and to reconstruct and repair any paving, grading, guttering and curbing, and to make and repair sidewalks, bridges, culverts and crosswalks, and to condemn and destroy any sidewalk deemed unfit for use, and replace the same with a new one of the same or different material, and to exercise control over streets and alleys, and establish and reestablish grades thereon. The cost of bridges, culverts and footwalks across streets and alleys shall be paid for out of the general revenue funds of the city. The cost of grading streets and alleys shall be charged against the lots and tracts of land fronting or abutting on the street or alley, or part of same, so improved, and on the improvement, in proportion to the number of fronting or abutting feet. Such city shall have the power to grade all, or any part of any street, or alley, but when the sidewalk part of any street, that is the part between the curb line and the street line, is graded exclusive of the other parts of same, the cost of the grading shall be charged against the lots and tracts of land fronting or abutting on the side of the street so improved, and on the improvement, in proportion to the number of fronting or abutting feet. The cost of making curbing and sidewalks shall be charged against the lots and tracts of land fronting or abutting on the improvement in proportion to the fronting or abutting feet, except that in making sidewalks, corner lots shall be charged with the cost of extending the sidewalk to the curb lines of intersecting streets, and in making curbing, corner lots shall be charged with the cost of extending the curbing to the curb lines of intersecting streets, and curbing and extending the curbing back to the street line at intersecting streets and alleys. The cost of repairing sidewalks and curbing shall be charged against the particular lot or tract of land fronting or abutting on the part repaired. The cost of paving, guttering and otherwise improving any alley and the roadway part of any street, that is, the part between curb lines, including street intersections, shall be charged against the lots and tracts of land fronting or abutting on the street or alley so improved along the distance improved, in proportion to the number of fronting or abutting feet. When the paving or guttering on any street or alley is only repaired (“repaired” as here used shall not include any improvement where the entire surface of a paving is renewed, but such renewal shall be considered as paving), the cost of such repairing shall be charged in the following manner, namely: The street or alley shall be divided into sections, a section being the distance from the center line of one cross or intersecting street to the center line of the next cross or intersecting street, and the cost of repairing each section shall be charged against the lots and tracts of land fronting or abutting on that section in proportion to the number of fronting or abutting feet.