Falling off the SA notch is a common result of trying to lighten the SA trigger pull. Most hammer guns depend on the trigger finger reflexes to work right. Take a SAA revolver. If the trigger pull of the full cock is too light, the finger will not be putting on enough pressure to pull the trigger past the half cock and safety notches, and the hammer will batter itself and/or ruin the trigger.
On that gun the SA pull was lightened too much. The finger pressure required for release was not enough to make sure the hammer strut (sear) cleared the trigger. That was fixed.
Now, the second problem, the "catch". With the sideplate off and the mainspring removed, take a look at the trigger and hammer engagement. The trigger should push the sear upward, raising the hammer, to a point. Then the bottom cam on the trigger should catch the bottom of the hammer and continue the movement. At that point, the hammer strut should be free of contact with the trigger.
I think the "catch" is at that point where the direct trigger to hammer contact takes over. It could be that the person who messed up the hammer also messed around with the trigger, or that the new hammer also has problems. One thing to check is if, after the trigger takes over, the strut (sear) becomes free with no further contact. If the strut is too long, it can interfere with that transfer, then snap forward, causing a "catch".
(That little action beween the hammer and trigger is the secret to S&W's DA trigger pull. As the spring compresses and pull weight increases, the force is shifted to a different leverage so the pull remains the same. Most DA revolvers have a problem when the spring stacks and the strut comes into line with the trigger so it is lying on the trigger, increasing the pull. S&W avoids that.)
Jim