The piano's control-lever is in its right-hand position when the Prof first hits the piano with the ax, and thus the unit is apparently already turned on, and so it should have been still playing; it would not have been silent and then suddenly resumed playing when the Prof hit it with the ax.
The paper roll is obviously meant for use in a multi-instrument calliope, since it has way too many parallel (side-by-side) "simultaneously playing" holes to be used in a piano, especially to play such simple "separate-noted" two-handed selections like these, without a lot of multiple-note chords or crescendos.
When the wooden crate first falls in the fountain in the master shot, it is clearly empty because it is floating. When the camera cuts to a closer shot, it has clearly been weighted down and is not floating.
The Prof heatedly turns the control-lever for the player piano to its left-hand or "off" position when he storms into his house, yet the lever is in the right-hand "on" position when The Boys are starting to carry the piano back out of the room and the Prof comes bellowing back into the room with the ax.
When the piano is chasing Ollie down the steps, the sloping side of the crate is to the right (as seen from the street). As it actually passes over him, the sloping side is suddenly to the left, then it's to the right again as it runs down the final flight.
When Oliver Hardy starts walking down the steps to see the policeman, and the piano crate starts sliding down the steps after him, the shadows of the camera and crew can be clearly seen on the side of the crate as it passes by them.
The paper music-roll that is loaded into the piano is labelled "Medley of Patriotic Songs", so it would not likely contain the first selection of "Arkansas Traveller", which is merely a comedic country-dance jig/reel and thus not really of the truly patriotic genre of "Dixe" and "Star Spangled Banner" which are also included on the roll.
The player piano is equipped with a large "on-off" control-lever that causes the unit to shut off when the lever is moved to the left. Yet while the piano is playing "The Star Spangled Banner", the Prof silences the piano by pushing its control-lever further to the right, which should not cause it to shut off.
According to the titles in the film the events take place later "that afternoon". When Stan Laurel and Oliver Hardy do eventually get the piano into the house the clock on the mantle reads 9:45. This cannot be accurate because it would not be daylight at this time of day.
When Oliver Hardy first falls in the water fountain and when Stan Laurel falls down into the fountain from the second floor, water marks are clearly visible on the walls caused from splashes during previous takes.
Obviously fake piano and empty packing-crate, since The Boys could never have lifted/moved either one so easily if the piano was real (two places in the film where this is especially obvious are when they first slide the box out --- quite effortlessly and noiselessly --- from their wagon, and when they casually lift and start to carry the piano out of the room when the ravingly-furious Prof orders them to take it away).
The professor's wife [horrified] enters the scene while he is smashing up the piano and explains its delivery was no mistake since she bought it for him for his birthday. A few seconds later, she says, "But I thought you hated pianos." Why would she buy one if she thought that?