Why should we abolish my coat? Wheres logic in that? And how, at the same time, does it magicaly can be mine PERSONAL, mine PRIVATE, and (in sugested future) a collectives property?

Nobody gives a fuck about your coat, do you honestly think that's the problem marxists have with private property? that someone might... rent out their coat? that's not the kind of thing we're trying to solve here, it's also something literally nobody does in the real world.

If you worked in a coat factory, and you make 100 coats a day, how much should you be paid for that? I believe profit is the stolen value of labor, so, the worker should make the value of 100 coats if they make 100 coats, that's the injustice we're trying to solve.

I own someones lawn and they clean my coat (barter exchange) - my coat is PERSONAL or PRIVATE? How does that differ if money involved?

I'd say that's personal, if you're paying them to clean your coat, i'd say they have a coat cleaning business and the coat cloaners should own that business... which it sounds like in this example they already do, so, nothing needs to change.

Now change the “coat” into the “factory” (a “garage”, a “hammer”, a “boat”), what’s the diference?

Whether you're one of the workers or not changes. If it's a coat factory, you just own the factory, and make money off the stolen labor value, while contributing nothing. In your examples, you actually are contributing, which makes you a worker, and someone who should get the full value of your labor.