ok, the slop in the front was lateral it appeared. ie, side to side only. it didnt appear to go up n down. there is a bit of side to side play, about the same you had. so as long as the bearings are in good shape, and the hub isnt warped out where the bearings go, then i say its normal. maybe a shim or two can releave the extra side to side play you have.
Spur Gear shaft. Normal as it is. once in the gearbox, you ll find one bearing only on the spur shaft. it will be installed on the left end of the shaft. the right side (where the shaft exits the gear box) just has a clip and rubber piece to keep dirt out. that is where the Perverbial Missing Spur Shaft Bearing goes. once you install the "missing bearing" the slop will be minimal if any at all. i was a bit surprised at the amount of play in the shaft in stock form. but once the gear box is installed and the spur plate (with bearing ) is installed the shaft is quite rigid and solid. but adding the Missing bearing is a good piece of mind thing to do imo. but the setup should be fine, once fully assembled without it.
the diff though looked to be Way to much slap. i would expect a siezed diff bearing or maybe the bushing is worn away inside the diff cup where the axle passes thru the case, or the diff gears (inside the diff) are shot and need to be replaced. i would lean to the last of these options and also suspect the diff housing is warped and some diff gears are toasted.
grab TRs diff saver cases and a set of gears ( HPI WILL warrenty them, typically with a new full DIFF!) and bearings and re do the diff. i have found no need for the HD Gears for the gear box, unless your looking to shave weight.
with TRs case installed on the diff, you can also start to over fill the diff ( more than just to the cross pins which is bout 1/2 full) to do a bit more tuning of the diff. currently im running stock 1000w oil, but next teardown im going back to 30,000 diff oil.