It seems what you mean by shirt in English depends on where you're from. At least in American English, shirt seems to be anything worn on the upper body, except jackets and underwear. So that will usually be tröja. When we say skjorta and shirt in this course though, we mean something that Americans would probably call a button-up, and if your wife calls that tröja, she is very unusual.
I'm an American and have always used the term dress shirt to distinguish it from a casual shirt. Check a general chain store like Macy's: Dress shirts are the second major category. https://www.macys.com/shop/featured/men-shirts?ss=true
It's a sound you don't have in English so obviously it's hard to hear at first. The TTS is actually pretty good here so it sounds right. Read more here: http://en.wikipedia.org/wiki/Sj-sound
Your ears will get much better at picking up Swedish sounds with practice. (but there will never be a good way of writing them with English letters, I'm afraid). :)