ja kładę miedzy przednim a tylnym siedzeniem. (I think, here should be biernik)
First of all, "ja" is definitely not needed and sounds strange in this place. It can and should be omitted.
And after "kładę" there should be a name of something which is being put, otherwise the sentence is still very strange :)
For example "Kładę torbę między przednim a tylnym siedzeniem". "Kładę torbę między przednie a tylne siedzenie" means exactly the same, and it even sounds more natural (although in official language I would rather see the first version). So it seems that you are right.
Here some examples: (why there are narzednik/instrumental?)
Ja kładę pod Twoimi stopami.
Duch doktora Mengele unosi się nad Wisłą.
kiedy idę przed swoim byłym.
idę przed pracą!
Where do you have all this stuff from? It seems to be not a very good source of knowledge.
Ja kładę pod Twoimi stopami.
Again - what is this "ja" for, and kładę what? What's more, it's rather hard to put something under somebody's feet, unless this person is leviatating :) There is an idiom "kłaść komuś kłody pod nogi" - literally "to put logs under somebody's legs/feet", meaning something like "to make problems to somebody intending to prevent this person from succeeding in what he does". Maybe it's the case.
OK would be e.g. "Kładę torbę przed twoimi stopami". "Twoimi" (and similiar words) is capitalised only in letters, only for the person to which the letter is going to be sent, and this sentence is rather unlikely to appear in a letter :) In this case "Kładę torbę przed twoje stopy" sounds rather rude (but all this sentence is still strange and a bit rude, in what a situation could it be used?). And I think it might be difficult a bit for foreigner to feel it, so maybe in this case narzędnik is just safer.
Duch doktora Mengele unosi się nad Wisłą.
Not towards the space above Vistula river ("nad Wisłę" would be OK only in that case), but just above the river, in one place, so there is no motion.
kiedy idę przed swoim byłym.
It means exactly: "When I am walking in front of my ex-husband.". As you can see, it's not a whole sentence :) And "Idę przed swojego byłego" would mean "I am going towards the place in front of my ex-husband", however strange it sounds (the original sentence is also very strange, I can't see any point in walking in front of your ex-husband).
idę przed pracą!
Is it from a computer translator? If "praca" is the building where somebody works/is employed there (I can't invent any other rational meaning of this sentence), it means "I am walking in front of this building" (I am there). "Idę przed pracę" - "I am walking to the place in front of this building" (I'm not there yet).
I hope it could you understand this difference :) And I think you had better change the materials for learning Polish :)