Yes, moving towards any object at speed in special relativity will make it seems contracted (shorter along the axis you're moving.). So the observer in the garage will see a contracted ladder. Meanwhile an observer seated on the ladder would see a non contracted ladder, but the garage would be contracted. The trouble of the paradox is of course that the garage does not get larger to allow in the "larger" non contracted ladder. This seemingly breaks what is called the principle of relativity, which is that the laws of physics are the same in all inertial frames of reference. The simultaneity they're discussing is the notion that events that seem simultaneous in one frame of reference, may not seem simultaneous in another frame of reference.

So where the first door would close and the second open at the same time to the garage observer. The second door would open much before the first door closes to the ladder observer.