How are some teachers, like her, qualified to teach?

Teacher supply shortage = any tom, dick, harry with a certificate can apply and most likely get the job.

Are adults easily offended?

Depends on the individual, I guess.

Is being quiet and an introvert offensive to others? Is being depressed and lonely equivalent to pointing a middle finger at others?

It's all about others' perception. You can say the most harmless thing in the world and a female colleague with irritable temperament on PMS will still take it the wrong way ¯\_(ツ)_/¯

Do I have to cope with people like her all the time after I graduate from school? Do I have to obey my teachers all the time because our grades are determined by them and they are older than us? Do I have to be a dog all the time in the future, licking the shoes of those who are more powerful than me?

Sadly, yes. This is Singapore, so unless you work in an MNC with AMDK majority, the workplace culture dictates that you need to "give face" to your boss, ie. never correct them in front of other colleagues.

That said, I still think your teacher is being rather unreasonable, if the story is true.