Am I correct in thinking that the 'Z' in 'Zeitung' is pronounced 'ts' as in 'the cats'?
no, Lehrer is with Artikel "der", we use "die" for plural. but Zeitung is usually with Artikel "die" when it becomes plural, we will say "die Zeitungen"
Die Lehrer = The Teachers Die Zeitung = The Newspaper
Should it not be a plural newspaper?
I was also confused by this. I don't quite understand why Zeitung is "female" and not just neuter (Das Zeitung). But my German girlfriend couldn't explain it to me either and said it is just one of those "things" - exceptions to the rules.
You're confusing sexual gender of real word objects with grammatical gender. They are (mostly) unrelated concepts. A newspaper is sexually neuter of course, but the word die Zeitung has feminine gender. One has nothing to do with the other.
It's not exceptions to the rules, it's the rule. And not just in German, for example Russian or French and many other languages have it the same way. In fact grammatical genders in Old English worked just like that too.
From what I'm deducting here, teacher is masculine but newspaper is feminine, that's why 'Die Lehrer' means 'the teachers' and 'Die Zeitung' means 'the newspaper'.