Here's a quick explanation for what those phrases mean:
-As soon as means "once" or "as fast as you can after a certain thing happens."
-As far as can talk about distance or length, and sort of means "until." If you were talking about reading a book, you could say "I only got as far as chapter 2," meaning "I only read until chapter 2." But we also have this phrase "As far as that is concerned," which is a tricky idiom and can mean a few different things.
-As long as means "if" or "provided that." For example, "X will happen only if/as long as/provided that y happens."
-As well as means "also" or "and". So, "I have to drive my kids to school and/as well as clean the house today."
1) As soon as the application of new project has been approved, please let me know. I need to reply to customer. (Because the person getting back to you depends on time, you use as soon as.)
2) I don‘t want to miss the basketball game, so I will run to the shower as soon as I get home. (This sounds correct!)
3) This data is all I got from customs. (Without context, you should use all. As far as doesn't work here, because it doesn't refer to length or distance. Also, data is usually singular - so, you would say "this data.")
4) I’ve been extremely busy recently. The new products in our company launch next month, so I have to prepare the presentation as well as two quality reports for company A. (Correct! The only changes would be that "launch's" should be "launch," and "next months" should be "next month" or "in the next few months.")