Does anyone know how much perpendicular load pressure can be placed on the shaft of this gearbox (link) before it starts having problems?
Minimum, if that. There is approximately 0.75" worth of output shaft inside the gearbox, which is NOT sufficient cantilever support for a 4" external shaft.
Notice the note about hardened carrier plates? That is because in the FRC 2007 season those plates would fail which would then cause the internal gears to bind up, even if the shaft was under minimal load. So regardless of your application I would high recommend you support the end of the shaft with some sort of mounted bearing plate.
Good god that 07 box was a nightmare if cantilevered. Definitely use a bearing plate at the end of the shaft with anything more than a minimal load. Otherwise its fun to have freshman take those things apart! 
None, really. A bearing plate on the end of it is pretty much mandatory or they’ll bind. With a bearing plate it’ll work fine in drivetrains / arms / whatever. My team direct drove our back wheels with this transmission this year using a similar setup to what people describe.
Support the end. No ifs, ands, or buts about it.
In 07, those things broke if you even looked at them wrong.
Andy, you nailed that on the money. Our Capitan had to take several gearboxes and make one that worked. Since most teams just chucked the gearboxes after they failed. BUT some teams did last all season without a failure (Some as in a very small amount). 
-RC
We lasted through 3 competitions and an offseason event with no problems, but then bound and broke in the second round of a scrimmage a few months after. We had the original bearing plate from banebots supporting the end, though we spent hours lining the thing up and it also had chain runs going from either side (so perhaps the side loads offset each other?).
So I took the gearbox home and rebuilt it, after about a $10 investment from banebots. Now it’s covered in grease, Vaseline, and wax and powers two 8" propellers in an underwater ROV that goes mostly in a straight line. Ya know, for my own experimentation
Depending on your application, on a five point scale, I would say somewhere between zero, zip, nada, nil and none.
EDIT: I tried to calculate a service factor based on *anticipated load/design load * but when I use *not much/0 *, I get an error. I may need a new calculator.
Team 801 back in 2007 used them to direct drive the center wheel of a 6 wheel dt with the end of the shaft supported we have not had a single carrier plate failure during 2 competitions and one off season event and a lot of driver practice.
Team 48 used four of this exact transmission in our modified linkage drive this past season. Each module included the optional Banebot encoder kit with the AM wheel cantelevered against the encoder case. The modules had post mounts on the top and bottom to allow for rotation. Two small cylinders connected the left and right module pairs, orienting the wheels either straight ahead or 90 degrees for a “transverse” mode.
With no way to avoid cantelevering (is that a word?) the wheels because the of the rotation, we too feared the side loads on the shafts knowing what happened to Banabot trannys back in '07. We fixed that by modifying the aluminum plates attached on the top and bottom of each transmission that held the post mounts, cylinder mounts, and rotational mechanical stops. We extended them over and past the encoder case. We then bolted an additional plate between the two plates against the encoder case with a hole around the shaft that contained a Delrin bushing. The bushing and plate supported the shaft and took the side load.
We went through 3 regionals, the Championship event, training a new drive team for IRI, IRI (including mentor competition), and several local demos with no transmission issues whatsoever. We were very happy with them.
Preferably, you would want 2 support points on either side of the load and no cantilevered force on the gearbox at all. You can make it work without support but it will break your gearbox.
-vivek
Perhaps I’m playing devils advocate here but, 816 used one of these transmissions in 2009 without supporting the end. But, we also cut down the shaft so it was only long enough to go through a piece of 1/4" Lexan and put a sprocket on.
I’d say depending on the application you might be able to get away without supporting the end of the shaft if it’s a light duty mechanism and you keep the load within a half inch of the gear box.
I realize that it might not always be convenient or even possible to use so many bearing supports but do try to get the cantilevered force as close to the face of the gearbox as possible and get at least one spare gearbox.
-vivek