To give credit, like with a business situation, is to give someone time/space to repay later. Let them buy something and they promise to repay you later. If you give me credit for learning Mandarin, it means to give me some leeway, some acknowledgement/ allow me some opportunity to show you what I can do.

It has different meanings depending on the context and "trust me" could be one of them, but probably not the most likely one. "Acknowledge that I've contributed" usually rebutting a presumption that I haven't. "Acknowledge that I have an ability." which could be similar to "trust me", also usually rebutting a presumption that I don't have it. "give me more points on my test" "allow me to borrow money" etc

Yes, it means put some trust in me as my past actions have shown I am trustworthy.

It means to understand that the person knows what they are saying or doing. Eg. He is a great actor, we need to give him credit. Eg. Give me some credit - I know what I’m doing!