I agree: you have done a great job! Many native speakers are very confused by these two words. I often hear people (even news readers on TV) say something like: There is a man LAYING in the street. I think that people would not make such mistakes if they remembered that "to lay" means something like "to put." So, of course, the man in the street is NOT laying / putting / placing anything. He is just LYING there.

I would change a little. I lay the book on the table everyday I already laid the book on the table I laid the book on the table yesterday The book lies on the table everyday (good job on the past two sentences!) The book laid on the table yesterday The book was laying on the table this morning If you have any question feel free to ask! (sorry I can't help you in German, my German friend is sick)

Those are all correct and usable!! ***Good job!