Does mind ever stop?

Mind has three parts - unconscious, subconscious and conscious. Unconscious mind is the animal part. It has to work to keep the body going. Subconscious and conscious part is where the problem lies. They are active most of the time and make you indulge in sensory world. Nirvana does not touch unconscious part otherwise you will dead in no time. Subconscious part is eliminated completely and unnecessary chattering of mind in forms of thoughts and emotions is stopped. Now you control the conscious mind because subconscious mind is no longer hitting it again and again. Your conscious mind becomes a tool for you to apply. Earlier prior to Nirvana you were a tool for conscious mind to play with.

So does mind ever stop?

When you attain Nirodha Samapahti and Phala Samapathi your mental process stops. In the case of Phala Samapathi the stop is only momentarily but the mental process come to a stop where you see the links of dependent origination breaking when 1st experiencing Nirvana and see the re formation of the links of dependent origination reforming when you come out. In other words when you experience Phala Samapathi you also experience Nirodha Samapahti momentarily at least. (Based on interpretation of S. N. Goenka and Banhte Vilimalasiri. These are highly theoretical and no consensus among different practitioners. Best is the practice and see for your self. Much of what is said about this is in secondary sources than from the Suttas themselves.) If you are stay in it you have to be Anagami or above having also reached the 8th Jhana.

'attainment of extinction' (S. XIV, 11), also called saññā-vedayita-nirodha, 'extinction of feeling and perception', is the temporary suspension of all consciousness and mental activity, following immediately upon the semi-conscious state called 'sphere of neither-perception-nor-non-perception' (s. jhāna, 8). The absolutely necessary pre-conditions to its attainment are said to be perfect mastery of all the 8 absorptions (jhāna), as well as the previous attainment of Anāgāmī or Arahantship (s. ariya-puggala).

According to Vis.M. XXIII, the entering into this state takes place in the following way: by means of mental tranquillity (samatha) and insight (vipassanā) one has to pass through all the 8 absorptions one after the other up to the sphere of neither-perception-nor-non-perception and then one has to bring this state to an end. If, namely, according to the Vis.M., the disciple (Anāgāmī or Arahat) passes through the absorption merely by means of tranquillity, i.e. concentration, he will only attain the sphere of neither-perception-nor-non-perception, and then come to a standstill; if, on the other hand, he proceeds only with insight, he will reach the fruition (phala) of Anāgāmī or Arahantship. He, however, who by means of both faculties has risen from absorption to absorption and, having made the necessary preparations, brings the sphere of neither-perception-nor-non-perception to an end, such a one reaches the state of extinction. Whilst the disciple is passing through the 8 absorptions, he each time emerges from the absorption attained, and regards with his insight all the mental phenomena constituting that special absorption, as impermanent, miserable and impersonal. Then he again enters the next higher absorption, and thus, after each absorption practising insight, he at last reaches the state of neither-perception-nor-non-perception, and thereafter the full extinction. This state, according to the Com., may last for 7 days or even longer. Immediately at the rising from this state, however, there arises in the Anāgāmī the fruition of Anāgāmiship (anāgāmi-phala), in the Arahat the fruition of Arahantship (arahatta-phala).

With regard to the difference existing between the monk abiding in this state of extinction on the one hand, and a dead person on the other hand, M 43 says: "In him who is dead, and whose life has come to an end, the bodily (in-and-out breathing), verbal (thought-conception and discursive thinking), and mental functions (s. sankhāra, 2) have become suspended and come to a standstill, life is exhausted, the vital heat extinguished, the faculties are destroyed. Also in the monk who has reached 'extinction of perception and feeling' (saññā-vedayita-nirodha), the bodily, verbal and mental functions have been suspended and come to a standstill, but life is not exhausted, the vital heat not extinguished, and the faculties are not destroyed." For details, see Vis.M. XXIII; for texts s. Path 206.

See: nirodha-samāpatti

12. Anuloma…if you compare it with a srewdriver and a screw, trying to break through a wall: 9-11 seems like to push and apply one’s whole force to the screwdriver and 13. anuloma is like turning the screwdriver to get through the wall. Its a mini-repetition of all nyanas prior to it, seeing a relentless series of udayabaya-nyanas, mini-nibbidas, mini-sankharupekkhas until the point of the screw peeks through on the other side. I guess others can find better similies…but in a way this sounds like fractals….the structure within the structure…creating a very very strong counter-drag to release the mind, and pull name&form from consciousness thus making the world (ie. 6 senses, ie. bhava or existence) tumble/stop for one moment – with everlasting consequences.

(Source: Vipassana Knowledge and the Path to Nibbana)

Also see: The Cognitive Process of Attainments, What is the different between Nirodha Samapatti and Phala Samapatti?, “Relief” with Bhante Vimalaramsi -Story about Dipa Ma

Does mind start with the birth and keeps going till the last minute?

See @sankha-kulathantille's answer

The mind occurs in both passive and active modes. The passive gives way to the active when a stimulus is received through one of the sense doors. The passive state of mind is called bhava"nga, cuti, or paa.tisandhi, according to the occasion.

Bhava"nga. The bhava"nga citta, mentioned earlier, is the primary form of mind. It flows from conception to death except when interrupted by a stimulus through one of the sense doors. Read more...

The stream of Bhavanga citta is what you have in deep(dreamless) sleep. It is a result of your past birth Karma. The process doesn't stop until the being attains Parinibbana, except in the case of being born in the Asaññasatta realm.