1. That sentence really doesn't make sense (with or without the 'a'), unless they're saying that there's pie that comes in a piece? Otherwise, yes your instinct is correct, adding the 'a' is better. 2. You could technically say 'the stand still' if there was more of a context, but really if you say 'the stand still' it sounds like a place. "Stand still" is most commonly used in sentences like: "They came to a stand still." and in sentences such as these you would never say the stand still. 3. Heir gets an article because it is a noun, and it is furthermore a very tangible noun. Meanwhile, their is a possessive adjective, and adjectives do not get articles.

There's only a slight difference between must and have to, and so the two can be used interchangeably. If you really want a detailed understanding of the two, then it is commonly said that 'have to' often expresses an obligation that comes from the outside, while 'must' expresses an obligation that comes from the speaker's feelings and wishes. For example: I must go on a diet, because I feel that I should. I have to go on a diet, because the doctor said so. Must you wear all black all the time? (implies that it is the personal choice of the person wearing all black) Do you have to wear all black all the time? (implies that there is a rule that the person must wear all black)

First notebook entry: Introducing myself

I am interested in learning English and hope to speak English well and to talk with foreigners freely.
Speaking English fluently will make me more confident and this is my goal, I believe that I can accomplish it and have a good career.

I wasn't sure if the last part is what you meant to say, hope it's alright! 

Cell phone is American slang for the portable/mobile phone. Generally in America, the phrase 'cell phone' is most commonly used (I rarely hear anyone say mobile phone unless maybe in a commercial), while apparently 'mobile phone' is used commonly in Europe.

There's really no difference between sum and amount, and so you could technically use either. Unfortunately one might sound more awkward than the other, depending on the context. I would say that sum has more of a mathematical implication to it, while amount is more general. For example, here they are in various contexts: The amount of money I spent yesterday was embarrassing. We divided the sum of money between the three of us.

My English class

I started my english class two weeks ago. I love it! My teacher is great, she lived in the US for two years and she got married there, but now she's single. I have five classmates, everybody is a girl and older than me. I want to learn English until I finish high school! Wish me good luck!!

First notebook entry: Introducing myself

I am very bad at English, but I would like to learn.

^^ That's harsh on yourself! Instead: I don't speak English very well, but I would like to learn.