Yes, men and women are both supposed to lower their gaze in modesty. And they are also supposed to cover their bodies (different restrictions for men and women though.)
The reason Muslim women, for instance, will cover is because they are commanded to in the Qur’an. Right after being told to lower their gaze… Why accept one command and not the other?
A lion you might see on a safari can run around but anyone can come by and watch. Muslim women don’t consider their bodies to be the property of anyone, that they be allowed to appreciate it without her permission. And they cover because God has told them to. Besides, what if a man decides not to lower her gaze–should he be allowed to see her because he doesn’t want to follow God’s command?
Now, you also asked about:
hijab vs. khimar vs. chador vs. niqab vs. burqa
These different words have different connotations in different contexts. What is prescribed in Islam is for a woman to cover her body in loose clothing which ought to obscure her shape. The exclusion is her face and hands, in general opinion. This is more or less the “bare minimum.”
The common way to describe dressing like that is with the word “hijab” which implies clothing that is full-coverage (including the head) and loose and not transparent. That is the meaning of hijab.
But, many people also use the word “hijab” to describe the headscarf exclusively. That is a common understanding these days, for girls to call their headscarves “hijabs.”
The word “khimar” is the word you see in the Qur’an actually, and specifically refers to a headcovering. In fact, it’s what many people mean when they say “hijab,” as in the Qur’an the women are told to take their headcovers and to cover their chests. So “khimar” is a headcover.
A “chador” is more like a cloak even, what they wear in Iran. This is wear you start to see different “styles” of covering in different parts of the world. Women are told in the Qur’an to wear an outer garment (why I said loose clothing that obscures the shape above), and a “chador” covers the head and body so it fits that requirement.
Now, “niqab” is what you see women in like Saudi Arabia wearing–in Riyadh I think women have to wear it actually. And there are some scholars in Islam who have the opinion that women need to cover their face (and even hands) in addition to everything else. And the “niqab” is a face-veil. There are many ways of wearing a face veil, but one way is with the black outer dress (jilbab, it’s called), a black headscarf, and a black piece of cloth tied around the face to cover it. There are other kinds of “niqab” but mostly it just means covering the face. Some people who wear chador might wear niqab as well.
The “burqa” is what you see in Afghanistan (and maybe other parts of South/Central Asia), part of the cultural dress there. It’s a way for women to follow these commands of covering their bodies in loose clothing. The “burqa” is worn on top of regular clothes covers the face as well with a screen through which the woman can see. It became mandated for all women to dress this way when the Taliban took control of Afghanistan.
So there’s a tradition in Islam of wearing loose clothes and covering the body, sometimes even covering the face. It’s just accomplished in different ways in different parts of the world, with different styles of dress.
I hope that answers at least part of your question.