Süstem und Tüpisch?
Is the letter Y really pronounced like Ü? Does "System" sound like "Süstem" and "typisch" like "tüpisch"? (I have heard this in some videos)
I don't know a lot of German (yet :D) but I was taught that the word Gymnasium is read the way you're mentioning. Gümnasium.
I am not sure about those two, tho :S
Yes, that is correct. Here is a nice little article explaining the pronunciation of the German letter Y in different positions within a word: http://joycep.myweb.port.ac.uk/pronounce/consony.html