Kaa states that cats do not like water; therefore, Shere won't follow Mowgli through water. However, tigers do like water, so it should not be surprising that Shere follows Mowgli when he jumps into the water.
When Mowgli goes to buy the knife and is accosted by Buldeo, the position and angle of the Buldeo's rifle changes between shots. When the camera is on Mowgli, the rifle is higher than Mowgli and points to his chest; when seen from behind Mowgli, the rifle is held lower and points more to Mowgli's belly.
Two scenes with the black panther were obviously shot with the panther behind a glass screen, likely as a safeguard to protect the actors. In both scenes, showing close-ups of the panther, debris is seen adhering to the glass.
When the thieves are entering the treasure chamber, the first thief sits beside a large granite rock and begins running his hands through the gold coins. As he does so, his knee bumps the rock and it moves, showing that it is clearly just a lightweight prop.