"Where do you live" is best. Then a person will reply with a city, country, etc. I would ask "where are you staying" only if the person is on vacation or at a temporary place. (Example - Question: Where are you staying? Answer: I am staying at The Grand Hotel (or) my friend's apartment As for the preposition, use "in" for live, and "at" for stay. (Example - I live in California. / I am staying at the hotel.)

I would say I am "okay" or "not bad." "So-so" works as well.

I started on the first paragraph... I've given you two versions. The first is yours simply corrected. The second has been enhanced for clarity.

 

Raising children has always been considered   the duty of women. That is why the idea that women are better parentS bearS (<em>the word "idea" is the subject so bear must be bears</em>)in people’s mindS. It is actually true everywhere in the world. In the last few decades, things HAVE tendED to improve over time thanks to modernization of THE couples’ ways of life, but things seem to remains the same: women are better parentS than men. (For proper writing in English, you should never start a sentence with "but" or "and", yet it is used that way everyday in conversation).

 

Raising children has historically been considered the duty of women. Therefore, the idea that women are better parents is still quite prevalent. It is true throughout the world. In the last few decades, there has been a modernation of the couple dynamic, yet the idea still pervades: women are better parents than men.

Looks good!

It depends on the context. Both could be correct.

Motivation is used much more often than impetus. I rarely hear impetus used in conversation. Motivation is usually something that comes from within; impetus is more of a general force.