What would a calculation be for the weight bearing capacity of 6 x 2 studwork?
The "deck" is probably fine assuming you used joist hangers, blocking, and is less than 10x10. Attaching it to the structure should be minimum 5\8ths through bolts. 

Follow this completely over blown code to eliminate any doubt. It is, however, for stuff you stand on, not climb.

http://www.cityofchicago.org/content/dam/city/depts/bldgs/general/Porch/Porch2011/RevPorch_Guide52611_Secured.pdf

I'd be worried about two things: pulling the whole thing off the wall and pulling off one of the campus boards, both solved with correct sized through bolts. Sounds like its solid as a rock, now you just have to make sure it doesn't come down. I think your landlord may be more concerned that it will fail from not being properly secured then any of the carpentry you built. For this you only need the specs from the fasteners used to secure it. Use 3\4in through bolts and washers where it attaches to the existing studs.

A grade 2, 3\4 bolt has a shear strength of about 10 tons.
https://nucor-fastener.com/Files/PDFs/TechDataSheets/TDS_013_Shear_Strength.pdf 
You should check the pull-out strength of the other fasteners you used and throw those numbers at him, too.

What did you use the silicone for? "fixed at 3 points into 8mm thick steel column" with what? Picture please.