In my essay I use the term "turn of the century".....

It seemed pretty “turn of the century” when you email your friends in the labor room.

Is this slang? Is it too informal?
Turn of the century in itself is not at all informal, Court. As an adjective, turn-of-the-century is also common: I have a fine turn-of-the-century jardiniere. If anything, the phrase seems rather over-formal in the casual sentence of your example; nor can I quite understand your meaning-- it seemed pretty old-fashioned?
I had the wrong impression of turn of the century. I changed it to high tech.

It seemed pretty “high tech” when you email your friends in the labor room.

Should I capitalize the H and mark it with "quotations or not. When including thoughts in a paper should I also use quotations marks?

” I thought to myself “Relax! How can I relax when you are planning a major operation on me that I have never had before? Relax, yeah right.”
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High tech and the adjective high-tech are quite common phrases now, so no need to enclose them in quotation marks or capitalize.

Yes, I would use quotation marks for thoughts.