This seems to be multiple questions within the broader theme. Some of which I’m not able to answer satisfactorily, others I hope I can elucidate a little on. Given the nature of your question I’ll focus more on the Imams that spent time in hiding rather than those that lived openly during their Imamates.

Firstly, one should keep in mind that there are multiple Imam Mahdis within different subgroups of Islam. I assume you’re referring to Muhammad ibn al-Hasan al-Mahdi the 12th and final Imam of the Ithna Asharis (Twelvers), who’s historicity I am not able to comment on. However, we can be certain that Abu Muhammad ‘Abd Allah al-Mahdi of the Ismaili tradition did exist (although the Ismaili conception of a Mahdi differs somewhat with the Ithna Asheris).

Secondly the tradition of a hidden Imam was somewhat precedented within Shia history within both a minor occultation (the Imam living in hiding) and major occultation (the disappearance of the Imam from the physical world) context before the major occultation of Imam Muhammad al-Mahdi of the Ithna Asheri tradition. The first appearance of a claim of major occultation within a Shia tradition was made by the Kaysanites, some of whom believed Muhammad ibn al-Hanifiyya (the fourth son of Ali ibn Abi Talib and contemporary of Ali ibn Husayn Zayn al-Abidin (the fourth Imam of the Ithna Asheris and the third of the Ismailis)) had gone into greater occultation. To further confuse things the Bayaniyya believed Muhammad ibn al-Hanifiyya had died and was succeeded by his son Abu Hashim. They then believe Abu Hashim didn’t die but went into greater occultation. The Nawusiyyah believed that Jafar al-Sadiq entered greater occultation rather than dying. Then the Fatiha who had two subsects over the greater occultation of Abdullah al-Aftah or his theorised secret son Muhammad ibn al-Aftah. The Qarmatians who believed in the greater occultation of Muhammad ibn Ismail, to say nothing of the competing people claiming to be each of these figures.

The tradition of hidden leadership through living in hiding (minor occultation) and communicating through community leaders was fairly common as a tool of revolution against Umayyad rule. This was generally only for an individual person however for the Ismailis spanned multiple generations of their Imams and is referred to as the dawr al-satr (period of concealment).

The first of these hidden Imams was Abd Allah ibn Muhammad (the eldest son of Muhammad ibn Ismail) who went into hiding to escape from Abbasid persecution and eventually settling in Salamiyya where he posed as a Hashimid merchant. He fathered two sons, with the eldest Ahmad succeeding him as Imam and then his son Hussein succeeding him as the final hidden Imam of that period. We know about each being related to one another as they are all buried in a family mausoleum and all their names are given by Ahmad ibn Muhammad in his “Istitar al-imam”.

As the previous paragraphs demonstrate following different lines of Shia Imams and the subsects within can be confusing. For example, the Imam you’re referring to at Alamut is Hasan ala-Zikris Salaam who claimed to be the Imam whilst the Fatimid Caliphate was still active under Abu Muhammad Abd Allah ibn Yusuf al-Adid li-Din Allah who also claimed to be the Imam.

To expand on how that came to be we need to go back to the succession of Abu Tamim Maad al-Mustansir Billah who all the then extant Ismailis recognised as the Imam. Al-Mustansir had designated his son Nizar to succeed him however the scheming of his vizir al-Afdal Shahanshah put another of al-Mustansir’s sons; Abdul Qasim al-Mustali on the Fatimid throne. Al-Mustali would also claim the Imamate. Nizar would attempt a revolution against his brother but would fail and be executed. Nizari Ismailis claim some of Nizar’s sons were snuck out of Egypt during or immediately following Nizar’s revolution. Those that followed Mustali later split again slightly after the death of Abu Ali al-Mansur ibn al-Mustali al-Amir bi-Akham Allah. With his infant son al-Tayyib Abul Qasim being his designated successor, but as al-Tayyib was so young his uncle Abul Maymun Abd al-Majid al-Hafiz ruled as regent. After a little over a year al-Hafiz declared himself Imam and Caliph and al-Tayyib who was quickly secreted out of Egypt, side-lined from the succession, or killed.

Hasan ala-Zikris Salaam claimed descent from Nizar through al-Hadi, al-Muhtadi and al Qahir. Unfortunately, not a great deal of information regarding any of these figures survived the Mongol destruction of the Nizari Ismaili State. What can be pieced together from the writings of Ata-Malik Juvayni, Rashid al-Din Fadl Allah and Abul Qasim Kashani show that the Ismailis at Alamut believed that a descendant of Nizar had reached Alamut (with the aid of a qadi called Abul Hasan Saidi) and was living in secret, only identifying himself to important dais (religious leaders) like the lords of Alamut. Hasan ala-Zikris Salaam became the fourth lord of Alamut and leader of the Nizari Ismailis in 1162 CE and was proclaimed the Imam in 1164 CE.

After the fall of the Nizari Ismaili State we know Shams al-Din Muhammad became the first post Alamut Imam. His Imamate lasted almost 50 years during which time the Ismailis briefly recaptured Alamut (before being put down by the Ilkhanids). Upon his death the Ismailis fractured again based on his succession between Muhammad Shah and Qasim Shah. Muhammad Shah’s line died out in the 18th Century CE, whereas the Qasim Shahi line continues until the present day.

Qasim Shah is relatively obscure but his son Islam Shah was a contemporary of Timur the Lame, who was documented to target Ismailis (though certainly not exclusively Ismailis) for pillaging and significantly worse in Anjudan.

The 32nd Ismaili Imam Muhammad bin Islam Shah lived posing as a Sufi shayk in Anjudan although he took a reginal name: Mustansir Billah, likely as a clearer way of associating with his Fatimid ancestry and missionary activity seemed to resume somewhat (although they continued to publicly deny being Ismaili). Mustansir Billah also likely established a connection with the Sufi Nimat Allahi order which had started to express more Shi’i leanings after the collapse of the Ilkhanids.

The advent of the Safavids during the Imamate of the 35th Imam Abu Dharr Ali initially seemed to spell the end of religious dissimulation for the Ismailis as the Twelver Shia kingdom seemed to tolerate its sister sect relatively well, with Abu Dharr Ali marrying into the royal line (either a sister or daughter of Shah Tahmasp I). This was not to last and the Safavids began to persecute the Ismailis roughly during the ascension of the 36th Imam Murad Mirza to the Imamate.

Murad Mirza for his part appeared to be aligning with the Nuqtavis. Seeing Murad Mirza’s popularity in Kashan and central Persia more generally Shah Tahmasp ordered his arrest in 1573 CE. After Murad Mirza’s escape from prison and subsequent recapture in 1574 CE he was executed and again the Ismailis were forced underground.

Murad Mirza’s son Khalil Allah, better known as Dhul-faqar Ali posed as Twelver publicly throughout his Imamate, even marrying a Safavid princess. He appeared to get on well with Shah Abbas, with Shah Abbas exempting the Shia community of Anjudan from certain taxes. Shah Abbas also appeared relatively indifferent to the Ismaili community so long as they continued their political quietism.

The 42nd Imam Hasan Ali settled in Kirman to allow a safer route for Indian pilgrims to visit him. He effectively ended the practise of dissimulation and openly proclaimed his Imamate.

The 44th Imam Abul Hasan Ali became the governor of Kirman under the Zands. After the death of Karim Khan, Abul Hasan Ali’s popularity allowed him to continue governing Kirman effectively independently. He supported Sadiq Khan’s claim to the Zand succession and after it was successfully pressed Sadiq Khan official appointed him governor of Kirman (again). Abul Hasan Ali would clash with Afghan, Balochs and the dregs of Zand claimants for control of Kirman until 1791 CE.

In 1792 CE Abul Hasan Ali would be succeeded by Shah Khalil Allah. The governorship of Kirman would fall to Abul Hasan Ali’s cousin Mirza Sidiq, this was subsequently conquered by Fath Ali Khan on behalf of Agha Muhammad Khan Qajar. The Ismailis immediately associated with Shah Khalil Allah were relocated to Kahak where they were given land and property as compensation for what they left in Kirman. Although Shah Khalil Allah would remain in Kahak until 1815 CE.

So, you can see that the post Alamut Imams are relatively well evidenced, I’ve omitted some of the less eventful Imamates here for brevity, but the other Imams are evidenced historically. The initial Alamut period presents the greatest challenge to evidence of the continued direct lineal descent of the Nizari Imams as it remains a gap in the historical record of the Nizari Ismailis. We know Nizar had children but to definitively conclude they became the progenitors of the Imams at Alamut is difficult.