you normally have to dismantle the gearbox to get it out and install it back since it needs pressing on the shaft and in the intermediate housing(if it slides easily then you have more problems), I would not recommend using puller without dismantling but it is possible. what is the problem with...
Most likely your 3rd/4th gear hub have fractured, its a common problem, can be repaired pm me.
(here same problem: https://itinerant-air-cooled.com/viewtopic.php?f=47&t=7320)
it is possible, but it all depends what is your gearbox code. fine tooth gears are only weaker if you have debris in your oil, they are smaller but there is more of them.