Thank you for your answers.
Unfortunately I do not meet the requirements for FAA ATP... Because of that I will have to take a more difficult path than
Chauderon did.
Part 61.41 says:
§61.41 Flight training received from flight instructors not certificated by the FAA.
(a) A person may credit flight training toward the requirements of a pilot certificate or rating issued under this part, if that person received the training from:
(1) A flight instructor of an Armed Force in a program for training military pilots of either—
(i) The United States; or
(ii) A foreign contracting State to the Convention on International Civil Aviation.
(2) A flight instructor who is authorized to give such training by the licensing authority of a foreign contracting State to the Convention on International Civil Aviation, and the flight training is given outside the United States.
(b) A flight instructor described in paragraph (a) of this section is only authorized to give endorsements to show training given.
Which indeed means I will basically also have to do the trainings all over again, not only checkrides and written tests. That is probably a sensible procedure for people with not so much experience (and none in FAA).
To sum up, the most suitable order of things in my situation would be to complete (after obtaining the 'based' private certificate and medical of course):
IR (training + written + checkride) -> Multi Engine (training + written + checkride) -> Commercial (same or similar) -> CFI (same) -> when I meet the hour requirements ATP-CTP, course, written, training and check ride.
Again please correct me if I'm wrong anywhere. Maybe there is another way of scheduling the trainings to reduce costs and time, or some of them may be combined together to do that, or maybe the order just makes no difference at all? Also, (regardless of how it could be done) as far as I understand there is no point in obtaining the TR that I have now on my EASA licence because I will just not be able to make any use of it.