The Hashashin were followers of Shi'a Islam, specifically the Nizari branch of Ismailism. The term they used to call themselves would more likely be Ismaili or the Nizari's. Ismailism is the second largest branch of Shi'a Islam, after the death of Jafar Sadiq (the 6th Shi'a Imam) there was a succession crisis as to who would be the next Imam, this led to the Shi'a schism and after that there was a schism in Ismailism itself (also involving a succession crisis) with the followers of Nizar would later be identified as "hashashin"). Hence it would be appropriate to say they would call themselves "Nizari" or "Ismaili".

As for the second part of the question, it is rather difficult to state whether they were involved in some "assassinations" or not. According to Marshall Hodgson in "The Secret Order of Assassins" he states that the order in Alamut led by Hasan-i Sabbah orchestrated various assassinations of high ranking administrators of the Seljuk, including Ahmad Sanjar's (Seljuk ruler of Khorasan) grand vizier. There are also other sources that claim they carried out assassinations for money; Abu Firas, an Ismaili writer, who wrote that he carried out an assassination ordered by Saladin.

The Nizari ismaili practiced "Taqiyya", which meant they did not reveal their true religion in public on the fear of being persecuted. Hence, many of their activities were conducted in secret which would help create legends regarding their activities.

My primary source, which I would highly recommend, is "The order of assassins: The struggle of the early Nizari Ismailis against the Islamic world" by Marshall Hodgson in which he described in great historical detail and accuracy about the history of the order, religion and politics.