if a beam of spin-up-along-vertical particles are put through a horizontally aligned stern-gerlach apparatus,

a left horizontal beam and a right horizontal beam are produced.

On either of these paths, if the vertical spin in measured, 50% of them will be found to be spin down-along-vertical.

If on the other hand, no measurement at all is made until after the point where the two paths are joined, none of them are found to be spin down along vertical at all.

question:

what happens if one path is made longer than the other but the paths are kept joined?

If the long path is made greater than c/t (where t is the measured transit time and c is the speed of light), then the particle couldn't possibly have come via the long path.

If no measurement at all is made until after the point where the two paths are joined, do they still all come out spin up-along-vertical in this case, even though the can be distinguished via the different transit times?

what if the long path is slightly shorter that c/t?

 

"c/t" has unit of "distance over time squared" so is an acceleeration,  not a distance.
 

How can the paths be 'kept joined' if a Stern-Gerlach splitter is introduced? The whole point of the SG experiment is to introduce a correlation between spin projection and linear momentum --which gives you the path.

Maybe there's something about your setup I don't understand. Perhaps you could make a drawing?