Sounds like you got a bum groupo. I mash on my Ultegra all the time, shifting while doing standing sprints, shifting chainrings and cogs at the same time, etc. and I'm not a dainty fellow by any means. I stress the heck out of the drivetrain. I also do my own adjustments and maintenance, so I keep a close eye on everything. With over 1500 miles on the bike this season (almost 1000 of them since the end of July), I've had to do only minor adjustments. If the bike is new, take it back and see what the shop'll do for you. None of those things should be happening.

Check your driveline angles and make sure everything lines up. Take it to a different shop and have the mech there look at it. My Shimano group shifts fine on my roadie.
On my P3 I have an issue with the Campy Record shifting from small to big ring. I have to spin easy before I can shift it back to the big ring. Going big to small is no problem. It was worse before until it was figured out that the braze on bracket was ever so slightly bent out of shape throwing the deraillure angle off. This caused the deraillure to be too much nose down. It would shift fine on the stand but under a load it wouldn't move off the small ring. I noticed that the back of the deraillure cage would flex when the chain was loaded. I shimmed the deraillure in the bracket and got the nose up more and after making the adjustments it shifts ok. After this seasons over I will look into replacing the braze on bracket.

From my limited experience, chains (especially shimano) rarely break.. Should malfunction of the chain occur, it would often be the result of an incompetent mechanic... I've equipped myself with basic maintenance knowledge and proper tools and am now competent enough to install a new chain..Yes, proper tools are expensive but it's worth the peace of mind knowing that your equipment was properly, and most importantly,lovingly installed by your own hands...:D