(a) This the city where I lived. (b) * This is the city I lived.
Only (a) is correct. For this sense of the verb “live”, either a relative word is required in adjunct function as in (a), or the preposition “in” must be added at the end and the relative word becomes optional: This is the city (that) I lived in.
Note that with a different verb like, say, “love”, the relative word is optional: This is the city (that) I loved, where the relative word (overt or covert) is not an adjunct but object of the relative clause.
It seems to bother the teacher that all the students are being too
The element in bold is not a relative clause, but a content clause functioning as extraposed subject. This is special kind of 'information packaging' construction called extraposition, where the subject of the basic non-extraposed version is moved (extraposed) to the end of the sentence, outside the verb phrase, and replaced with the dummy pronoun “it” which becomes the subject.
The non-extraposed version would be:
That all the students are being too quiet seems to bother the teacher.
The extraposed version would be more likely than the basic one in examples like this, and hence considered to be the default construction.