You mentioned something that was not revealed before about the old engine, (BROKEN CRANKSHAFT) where did the crank break???? The 93-95 engines are known for the crankshaft snout to break. (They usually break in the area of the harmonic balancer where the timing belt cranshaft gear is on the crank) The 93-95 crankshaft have a smaller snout than the 96's and newer. The main reason these cranks break on these engines is that the drive belts (PS, alternater, and a/c belts) are OVERTIGHTENED causing excessive pressure on the crank causing it to break. Make sure that the belts are not overtightened on the engine that has been installed.
Take the time if you can and inspect the old engine yourself to see where the crankshaft broke. If indeed the crank did break it would act as if the timing belt broke. If you friend has air wrenches it will be easy to remove the harmonic balancer bolt and the balancer should come off crank with little pressure. ANother thing you could do is try moving the crankshaft at the harmonic balancer to see if it has a movement verifying the broken crank.
Did your friend install a new timing belt and water pump in the used engine that you had installed as it would have been the ideal time to do it. When the engine was out of the vehicle it would take a fraction of the time to install??? I installed a new timing belt in my 94 about a year ago and it took me about 8 hrs to do from start to finish. (Engine in the vehicle)
Your friend removing the original engine and installing the salvage yard engine for $300.00 was a very fair rate for the job.
I would consider replacing the distributor because you say it has shaft play, so in time it is only going to get worse and leave you or your family stranded in the future. Find a good used one as they are exspensive new.
Personally I would inspect the old engine so you know the complete story on it before talking with family to prevent any hard feelings.