What is the status of the hadith of the Prophet (s), that is being circulated online, that says 'there will be 12 Mahdi's after the 12 Imams'? Does it exist in any book, and is it authentic?
Yes, the hadith exists. However, some people use it to promote movements or groups that are active today, and that is not correct. The hadith is speaking about the time AFTER the reappearance of the 12th Imam (i.e. the Mahdi), an event that will be accompanied by major signs. We will know when it happens, and it has not happened yet!
The hadith is mentioned in Bihar al-Anwar, vol. 53, p. 148, no. 6 and 7. One says that after the Mahdi (A) appears, the other Imams will be brought back to life and live and die, and then after them, 12 "mahdis". The other just speaks of 12 "mahdis" after the "mahdi".
Since this hadith is about something predicted for the end of time, it isn't possible to say whether it is correct or not. There is consensus on certain general predictions for the end of time - such as the return of the Mahdi, the raj'ah or return to life of those of most intense faith and most intense wickedness, and some major signs. However, details like this may or may not be accurate.