I'm so sorry you have to deal with this ugliness. But, I must admit that your story-telling ability has me in tears from giggling so hard. If nothing else, your frustration has given someone else some joy.
(My favorite is that you "intimidate said machine". Picture-perfect.)
Hope this resolves more quickly than you anticipate.
Oh, it's not that ugly, it's just a complete farce.

I'm honestly not half as stressed as I sound about it anymore*, but the whole situation's just too absurd to resist the poking of the funs.
Anyway, I figure by the end of the week we might actually be through a third of the letters.
*There was stress. Then some of my coworkers said (minus paraphrasing and artistic license), "Stop stressing! The whole thing is stupid, you know it, we know it, and CompanyCo knows it. Now rein in your work ethic before you go crazy and take the rest of us with you!"
Awww, the poor neglected little paper-folding machine just needed some attention and training. Is it getting praise and ear skritches--I mean, umm, pats? (What would be analogous to ears and ear skritching on a paper-folding machine?)--now that it's trying to please? 
As an expert in this area, I used to have this relationship with the postage machine when I worked for BMW, the equivalent to skritches is to praise the machine for it's hard work and discipline while in the earshot of others. This pleases the machine, and inspires it to future cooperation. Never neglect this integral part of the machine-human relationship, or you will find yourselves in the mines when the revolution comes.
Honestly, even with "training", I'm still going to be harboring a deep desire to go Office Space on its mechanical butt before this is all through. It's not so much trying to please as bending reluctantly to my adamantium will. Bow before me, puny paper folder!
Just so you know , this actually qualifies as a SuperPower . Now you must acquire Lightning Speed , X-Ray Vision and the Ability To Fly .
Then you gotta pick out a cool Super Hero name .
Well, as stated above, I have Adamantium Willpower, so that must count for something.

Alas, I get vertigo, so as much as I'd love to fly I'd have all the natural grace of a june bug.
Just so's you know, this did prompt a discussion between myself and my DH of him being a sidekick and wearing hot pants. His response, "Oh, you'd LIKE that, wouldn't you?"

And so that I don't leave you all with the impression that my workplace is terrible, I really do have great managers and coworkers. I got an official recognition for being able to tackle all of the things that've come my way (including jumping into two other programs, because we're just that short-handed and I'm flexible and learn really quickly) and Generally Being Awesome (tm) today, so that made me feel pretty good. And now that I'm not a temp anymore, I get
bennies!