Hi Samar! The "le" and "ne" both have different usage in different situations. It depand on what kind of meaning you want to express. In the article, I just list the usage for sentences there. For example, "le" can be used to make a question sound more friendly, so you can add it after any question, but you don't have to. Also it can be used after a noun or pronoun, which means what about ...? Like in the typical conversation: --Nǐ hǎo ma? --Wǒ hěn hǎo, nǐ ne? In this situation, you have to use it. 

If you want to know more about these two particles, you need to check a grammar book. For "le", I have found the webpage which explains it very clear:http://resources.allsetlearning.com/chinese/grammar/Uses_of_%22le%22 

Hi, Olya! If you say 我请你吃饭 to somebody, it means you will definitely pay.

You are right. If you mean just everyone pays for themselves or you share the fee together, you can say “我们去吃饭吧，行吗？”