Double clutching and the presence synchromesh has nothing to do with with whether the gears are helical or straight cut.
All modern transmissions have pre-meshed gears, the gear selector engages and disengages (dog)clutches to choose which pre-meshed gearset to use. In a non-syncro transmission if the the gearset side of the clutch is travelling faster than the selector side of the clutch, you'll hear grinding as you try and engage.
Synchromesh is a friction cone that gradually engages the two halfs of the clutch in an attempt to bring the selector clutch half to the same speed as the gearset clutch half. If the speeds of the two halves are not closely matched it can take great effort on the gear selector to engage the desired gearset as the synchromesh cones are required to do more work. This is often encountered when trying to select first gear while the vehicle is moving, or if while at speed the engine was allowed to idle with the clutch out in neutral before selecting another gear. In these cases and others, it is still beneficial to double clutch with a synchro mesh transmission, to ease the effort required to select the gear, and to reduce wear on the synchromesh cones.
Synchromesh can also wear out, not that it matters in GT5.