There is some wear on the 2-3 shift fork, it could cause some of the problems you are experiencing. The "big gear with the thin collar" is actually the synchronizer assembly. It is also referred to as a "drum". It has brass rings on both ends which help engage 2nd & 3rd gear.
It is engaged by the 2-3 shift fork moving the collar. 3rd gear is to the front. Take a close look at the ends to see if the brass rings move when it is moved manually. If the rings are loose or worn excessively it won't engage or disengage properly.
Fred
