factory ones are viscous, i had my gearbox overhauled some time ago, and mine has got a Mechanical LSD in it now i remember the mechanic telling me it didnt need any special stub axle, its been a while so i dont remeber what was said.
but recoing a box will cost around 1.5k, the 2nd and 3rd synchros are very notorious for going on mr2 so i would def fix them in whichever of the box you choose to go with.
if the box has never been opened before and they say its got some kind of lsd then it has to be a viscous one.
if your going with viscous one its not a repairable unit and once its worn out it has to be replaced. mechanical clutch type ones can be serviced, problem with the clutch type ones are they never are quiet and will shudder a lot if the wrong oil is used in the gearbox, use the right oil for the LSD and the synchros will start playing up and get hard to engage, its a no win situation, so you have to pick one or the other(smooth shifting or LSD chatter) you can get them to a happy medium but involves a lot of headache trying to get the right amount of LSD additive into the Gearbox oil so that it still shifts ok but also doesnt make the LSD chatter. i still havent found that medium yet cause i gave up lol.
cheers