2) 華語 / 中文 / 漢語 / 普通話 / 國語

These terms basically mean the same thing (as 'Mandarin Chinese language' in English) but a subtle difference is there in Mandarin Chinese regarding PeOPle spOkEN tO.

華語 is in comparison of 英語, 俄語, 阿拉伯語, etc.
Hence you would ask an Asian-faced lad in Singapore, ' Can you speak 華語? As I know, not every of you speak that language, though it's in your official language list. '

漢語 is in comparison of 藏語, 壯語, 苗語, etc.
Hence you would ask a Hmong gal (not Chinese, ethnically speaking, but with PRC citizenship) from Guangxi, ' Is 漢語 your first language? Or 苗語 is? '

普通話 is in comparison of 廣東話, 四川話, 上海話, etc.
Hence you would ask a man from Guangdong (Chinese, ethnically speaking, but not necessarily speaking Mandarin as a first language/dialect at home), ' Do you speak 普通話 well? '

國語 simply means 'National Language' by literal. Mainly called by Taiwanese people, Half-Taiwanese people or Taiwanese-related non-Taiwanese people.
Hence you would ask a Taiwanese boss running a tea shop in California, ' Do you mind teaching me 國語? '

中文 basically works as 華語 does, but more emphasis on a written way.
Hence you would ask a guy fluent in speaking Mandarin, ' Wow your speaking is just like no other, but can you write 中文 that well too? Well since I have met some speaking excellent Arabic yet they don't write. '

By the way, a lot of people learning Mandarin Chinese mistakenly write or say 中國歌手 (literal: singer from China) to mean 'Chinese singer' while 華人歌手 (literal: singer with Chinese ethnic background) or 華語歌手 (literal: singer who sings songs in Mandarin Chinese language) should be the proper way. It's not right to call a singer singing Chinese language 中國歌手 without considering if she or he is from Mainland China or not when she or he is actually from Malaysia / Taiwan / USA / Hong Kong / etc.

我的新鄰居

上個月，我在还沒上班之前, 我遇到了我的新鄰居。我聽見她正在對她兩個小孩講韓語。因此所以我以為她是韓國人。然後我跟她說一声，『안녕하세요』。她轉头回来向我，臉上看起來很驚奇訝 [wonderment, instead of surprised in fright] 。現在每當遇到她的時候，我都會跟她打招呼。這個星期日，我看到見她和她老公指著我，所以我走過去，向他們介紹我自己。我問他們，你們都來自韓國嗎？她老公說，『我來自韓國，我太太來自中國。我說，『真的嗎？我立刻感覺很開心。然後，我跟他太太說，』妳會說中文嗎？她回答道，『是，是，是的』。她不停地笑。我們都用中文普通話對話聊了十分鐘。睡車內的他們小孩在嬰兒車裡午睡覺醒來了以後，我們的聊談話就結束了。不過但是，她老公告訴我，“我太太正在試著學習英語"。然後，我說回答 [This is how the Chinese mind works in letting you know they needed help without causing embarassment if you had said no]，“如果將來妳她需要[Correct use instead of 須要!!!] 有人幫助妳來學習英語的話，可以隨時來找我” 然後而 [Too many 然後]她的老公對我說, "她可以教你中文漢語。你也可以教她英語."
“對了！ 大家互相學習”,  告別後 [Since you are using the full form.  The simplified actually means "queen"]后，我們都很高興地走開分手了。

Your Chinese is very good.  Except for some cultural nuances, I am very impressed by the length of your essay.  I am very sure in a very short time, will master the language.  I think it is time for you to delve a little deeper into the cultural and traditional aspects because in order to fully master a language, you need all those accoutrements otherwise your language has no soul in it.

Last month , while on my way to work, I met my new neighbor . She spoke to her two two children in Korean . So naturally, I thought she was Korean . So I said, " 안녕하세요 ." She turned to me, astonished. Now every time I met her, I say hello in Korean. This Sunday, I see her talking to her husband and pointing at me, so I approached them and introduced myself. I asked if they were from Korea ? Her husband said, "I'm from South Korea; my wife is from China."

I said, " Really? Your wife is from China?!!" I immediately felt very estatic. Then I ask her In Chinese, "Can you speak Chinese?" She replied, "yes, yes, yes!" She could not contain her happiness after learning I could speak a little Chinese. We had a ten-minute conversation in Chinese . Their children who were asleep in the car woke up from their nap. Before our conversation ended, her husband told me, " My wife is trying to learn English" to which I responded in Chinese," If in the future you need someone to help you learn English , you can always come to me." Then her husband said, " She can teach you Chinese. You teach her English ."
"Yes ! Learn from each other," she said. After we bid farewell, we all walked away happily.

我的新鄰居

上個月，我在上班之前遇到了我的新鄰居。我聽見她正在對着她的兩個小孩講韓語。所以我以為她是韓國人。然後我跟她說了『안녕하세요』。她轉向我，看起來很驚訝。現在每當遇到她的時候，我都會跟她打招呼。上个星期日，我看見她和她老公指著我，所以我走過去，向他們介紹了我自己。我問他們，你們都來自韓國嗎？她老公說，『我來自韓國，我太太來自中國。我說，『真的嗎？我立刻感覺很開心。然後，我跟他太太說，』妳會說中文嗎？她回答道，『是，是，是的。她不停地笑。我們用中文聊了十多分鐘。等睡在車內的小孩醒來以後，我們的談話就結束了。但是，她老公告訴我，“我太太正在試著學習英語。然後，我說，“如果妳需要有人幫助妳學習英語的話，可以隨時來找我。”然後她的老公說, "她可以教你漢語。你教她英語."
“對！ 互相學習。” 告別后，我們都高興地走開了。

Last month , while on my way to work, I met my new neighbor . She spoke to her two two children in Korean . So naturally, I thought she was Korean . So I said, " 안녕하세요 ." She turned to me, astonished. Now every time I met her, I say hello in Korean. This Sunday, I see her talking to her husband and pointing at me, so I approached them and introduced myself. I asked if they were from Korea ? Her husband said, "I'm from South Korea; my wife is from China."

I said, " Really? Your wife is from China?!!" I immediately felt very estatic. Then I ask her In Chinese, "Can you speak Chinese?" She replied, "yes, yes, yes!" She could not contain her happiness after learning I could speak a little Chinese. We had a ten-minute conversation in Chinese . Their children who were asleep in the car woke up from their nap. Before our conversation ended, her husband told me, " My wife is trying to learn English" to which I responded in Chinese," If in the future you need someone to help you learn English , you can always come to me." Then her husband said, " She can teach you Chinese. You teach her English ."
"Yes ! Learn from each other," she said. After we bid farewell, we all walked away happily.

我的新鄰居

上個月，我在上班之前遇到了我的新鄰居。我聽見她正在對她兩個小孩講韓語。所以我以為她是韓國人。然後我跟她說，『안녕하세요』。她轉向我，看起來很驚訝。現在每當遇到她的時候，我都會跟她打招呼。這個星期日，我看見她和她老公指著我，所以我走過去，向他們介紹我自己。我問他們，你們都來自韓國嗎？她老公說，『我來自韓國，我太太來自中國。我說，『真的嗎？我立刻感覺很開心。然後，我跟他太太說，』妳會說中文嗎？她回答道，『是，是，是的。她不停地笑。我們都用中文對話了十分鐘。睡車內的他們小孩午覺醒來以後，我們的談話就結束了。但是，她老公告訴我，“我太太正在試著學習英語。然後，我說，“如果將來妳需要有人幫助妳學習英語的話，可以隨時來找我。”然後她的老公說, "她可以教你漢語。你教她英語."
“對！ 互相學習。” 告別后，我們都高興地走開了。
This sunday,   I think it is last sunday 上个星期日

DaWei , I really really feel amazing to your efficiency learn Chinese .your Chinese level improved Is really so fast. this essay almost have not mistake.  I think you must find a good teacher. and you really a good learned man，I need to learn to you.