I totally agree with Andriy - your version is the correct one. The original one doesn't make much sense!

Your version is correct. There is no need for past perfect here - the actions are simultanious

Both answers are right: "As soon as he got into the car, I said good morning to him" is correct and is more commonly used . In order to say "As soon as he had got into the car, I said good morning to him", you would need to change it a little bit to "As soon as he had gotten into the car, I said good morning to him."

There is perhaps a difference between British and American English here. British English doesn't use "gotten" and the original version ("As soon as he had got...") would be considered correct, especially in speech. It also puts the action a little further back into the past and implies that he has climbed in and settled himself before you wish him good morning. "As soon as he got into..." makes the whole action more urgent. The greeting comes while he's still in the process of climbing in -- at least that's how I'd read it in a story.

The only way to bring a past action into present relevance is by using the perfect aspect. That said, "As soon as he had GOTTEN into his car, I said good morning to him" is the correct (ie, standard, proper English) sentence. In conversational English, "As soon as he got into..." would be acceptable. It all comes down to whether or not you want to speak in standard or conversational English. Hope this helps you understand better! :)