A Short Life of Sri Ramakrishna by Swami Tejasananda (published by Advaita Asrama) states that he used to consume large amounts of ice during the summer period of 1885 which led to a badly affected throat and eventually he was diagnosed with throat cancer shortly thereafter.
So, if we are to link the disease to any direct cause then that cause will be ice consumption and not that of betel leaves/nuts.
However, there is another story according to which Sri Ramakrishna used to allow all sorts of people touching him when they came for his visit. And, among which a lot of people would be mortal sinners. Their sins transmitted and assumed the form of a deadly disease.
The Mother Goddess once showed to Ramakrishna scars in his throat and explained how those came. This was much before he actually was affected with throat cancer.
During the sultry months of the year 1885 the Master suffered a great
deal from the terrible heat, which was alleviated by the taking of ice
the devotees brought to him daily.
But this frequent use of ice produced a pain in the throat which was
at first so trifling as to pass unnoticed. But in a month it became so
aggravated that the disciples grew alarmed and called in a specialist
in diseases of the throat. Notwithstanding the prescription of various
remedies, his condition did not show any sign of improvement. On days
of the full and the new moon, the pain became more acute; it was
impossible for him to eat solid food. The doctors diagnosed the
disease as ‘Clergyman’s sore throat’. The Master carried out the
doctors’ injunctions faithfully, except on two points, namely checking
the divine ecstasies and giving rest to the vocal organs.
And, it was in June 1885 when he was diagnosed with throat cancer as mentioned in the book "A Short Life of the Holy Mother".
In June 1885, Sri Ramakrishna developed cancer of the throat, and this necessitated his removal in October to Shyampukur in Calcutta for
treatment. The devotees arranged for everything regarding medical
care, nursing, and so on.
Second part of the question, did Sri. Ramakrishna Paramhansa died in
samadhi like his disciple Swami Vivekananda and was he aware of his
death time just like Vivekanada was??
It is mentioned in the same biography I have quoted from that he entered Mahasamadhi. The full details are as follows:
At last the eventful day arrived—a day of intense grief for the
devotees. It was Sunday, August 15, 1886, the last day of Shravana.
The Master’s suffering was at its highest. The devotees wept in grief.
They stood by the bed-side of the Master. In the evening he suddenly
fell into Samadhi. The body became stiff. There was something about
this Samadhi which struck Shashi as unusual, and he began to weep.
After midnight Sri Ramakrishna regained consciousness. The Master in a
clear voice uttered thrice the name of Kali and gently lay down.
Suddenly at two minutes past one, a thrill passed through the Master’s
body, making the hair stand on end. The eyes became fixed on the tip
of the nose and the face was lit up with a smile. The Master entered
into Mahasamadhi. Thus in the early hours of Monday, the 16th of
August 1886, Sri Ramakrishna departed from the world, leaving behind a
host of grief-stricken devotees and admirers.