Corrections in red.  Comments and stylistic suggestions in blue.

Articles. They are killing me >.<

Hello everyone! How is it going? It's spring here, and my Siberia is getting warmer and nicer :P
Despite my nice mood asking me to walk under the sun and enjoy the warmth, I keep working on my English as hard as I can. I still don't understand articles and have no idea which one I should use in many cases. For example, I didn't know what article I have to use in the sentences below:
- We live in a small appartment near the centre of the city.  Correct! 
- This morning I had a boiled egg and toast for breakfast.    Correct!
- I saw a $1 note on the pavement.     ($1 bill in American English.)
- I went into the shop and asked to speak to the manager. (I have no idea why I used "the" at first and then "a", I just feel like that).   If the shop had already been identified and everyone knows which shop you're talking about, you say "the".  If it's the first mention of the shop, and no one knows what shop you mean, then say "a".   A "shop" usually has just one manager, so "the" is better here.
- Did the police find the person who stole your cell-phone?   Correct!
- They went out for a meal last night.   Correct!
- Could you close the door, please? (Please, tell me that "the" is correct here :D)   Correct!
Ugh, it seems like I will never be able to understand this. Though I am not going to give up, but will keep making mistakes as long as it's needed in order to understand this stuff.

Very well written.