it is normal to have rotational backlash in these manual transmissions. There is no adjustment for the gear lash, but there are shims that can tighten the bearings, but that only affects the endplay.
For transmissions PRIOR to 2005 you can pull the yoke and see if the tail shaft has side-to-side play which means loose bearings. Probably up at the input shaft, but to do that test on a 2005+ manual transmission you will have to remove the big flange to see if the output shaft is loose.
In most cases when I hear symptoms like this, the noise happens as soon as the gear is engaged, but the clutch is still pressed to the floor. This noise should not happen in that condition even with a bad u-joint because the gears in the transmission should not be trying to turn.
I believe this to be a condition where the CLUTCH is not fully releasing for some reason.
Now, for an automatic transmission there will always be some rotational energy when engaging a gear and what I am saying does not apply.
Stan