It's VERY, VERY strange that you never had any problems with the Medium helicopter since I can Reproduce these issues something like 100% of times!
Are you sure that you're playing on "Expert" dificulty or that you played enough time with the Medium helicopter (more than a simple "5 minute trip").
I always had this problem with this medium helicopter but in previous version if I was very carefull with the helicopter (treat the helicopter like it was actually "made of glass") I could play and end missions with the medium helicopter without too many extensive strutural damage. But now with version 1.04 things are MUCH WORSE, even if you treat the medium helicopter "gently" it will inevitably get damage by itself specially loosing tail rotor authority which is a thing that didn't happen (tail rotor "stuff") in previous version 1.02 for example.
Just try the following:
-> With patch 1.04 installed play a "lift cargo" single player scenario/challenge with the medium helicopter and tell me if you can end that challenge without loosing the tail rotor and at least end it without extensive "strutural" damage!!
Honestly this is so easily reproducible since with the "light" and "heavy" helicopters you don't get that kind of exagerated and unrealitic damage like you have with the medium helicopter!
I've done several missions with the heavy helicopter. I have never had a problem with it, even on "Shakedown" where you need to do a few edge-of-the-envelope type manuevers. I've only ever played expert. Try a large amount of curve on your rudder. I have found that both the heavy and the medium are very easy to corkscrew if I am not careful with the yaw rate.
[Edit: I did try the mission where you sling 3 boxes from one location to another and had no problems.]