As of now, I haven't been able to find a proper reference on how the life of Maharishi Vishwamitra, who was Sri Rama's Guru, ended on this Earth. Do any authentic scriptures mention about his life's complete history and when it ended?
There is no such mention in the scriptures that Vishwamitra has ended his life. On the contrary, he is mentioned in several texts as one of the Saptarshis (seven sages) for this manvantara.
In the present vaivasvata manvantara, the seven great sages are Atri, Vashishtha, Kashyapa, Goutama, Bharadvaja, Vishvamitra and Jamadagni. The gods now are the sadhyas, the rudras, the vishvadevas, the vasus, the maruts, the adityas and the two ashvinis. From Chapter 3, Manvantaras, Brahma Purana
Vishwamitra is also mentioned asone of the saptarshis in other texts as well.
kaśyapo ’trir vasiṣṭhaś ca
viśvāmitro ’tha gautamaḥ jamadagnir bharadvāja
iti saptarṣayaḥ smṛtāḥ Srimad Bhagavatam 8.13.5
Kaśyapa, Atri, Vasiṣṭha, Viśvāmitra, Gautama, Jamadagni and Bharadvāja are known as the seven sages.
Sapta rishis do not leave this world after this yuga or manvantara. The sages and other beings reach God at the end of the Kalpa and are created again )
All beings, O Arjuna, enter into My Prakrti at the end of a cycle of time. Again I send these forth at the beginning of a cycle of time. (Bhagavad Gita 9.7 Translated by Adidevananda)
However, there are some exceptions like chiranjeevis. We do not know if Vishwamitra is a chiranjeevi. So, we can conclude Vishwamitra is still alive and will remain till the end of this kalpa.