I will try to explain "should" for you. "I told her, she SHOULD go to the store" = I suggested it, but she might not... "I SHOULD quit smoking" = but I probably wont... "They SHOULD teach more about other cultures in school" = but they don't... This is a suggestion. hence; I am talking to someone: "You should buy a car, Amy." I'm telling someone else later: "I suggested that Amy buy a car". or "I suggested Amy should buy a car" Does that make sense?

We can't say*

Thank you guys! According to my text book( It's English grammar in use) we don't say i suggested her to buy ;)

Should can also suggest that the one being spoken though really has to do something (but not in a forcing manner per se). Example: You should watch that movie! -> You really have to watch that movie! (because it's so good!) You shouldn't do that! -> Stop doing that! But as you said, it's quite profound and has multiple ways to be used. I don't understand the third sentence, I think the correct way to say this would be: "I suggested her to buy a car.", which is just normal past tense.

Numbers 1 & 2 are good. As far as number 3, I'm not saying it's wrong, but I cannot think of any situation where you should use it.