No human being asked Allah for Paradise (Jannah). Allah forcefully put them in a severe test. Is it right to punish who fail the test? How can he be RAHMAN in this case? He is so angry with polythiesm (shirk) that He can put a soul in the extreme torture, how He can be Samad? Many ahadith in Bukhari say that Paradise (Jannah) and Hell (Jahannum) is written on Loh for every soul. A man may do bad deeds for life and at the end he will make toba and go to Jannah as per his naseeb and vice versa. What remains for the test then? And where is mercy (Rahma)?
Answers to your questions according to my understanding are as follows, the numbers in the brackets indicate the Sura and the verse number of the Qur'an that are related to an argument:
1. No human being asked Allah for Jannah. Allah forcefully put them in a severe test. Is it right to punish who fail the test?
The above question is based on a series of assumptions and understandings that to me are not true or accurate:
A. Human being, as we know it, is defined as an entity that was made to be challenged in order to make himself good enough to be as close as possible to his Lord in Jannah by the way of being his perfect servant (51:56). Therefore to say no human being asked God for Jannah is not a self consistent statement. This is like saying that no mobile phone asked to be a tool for communication. If something is not a tool for communication then by definition it cannot be called a mobile phone. Likewise an entity that is not for the purpose of being challenged to be put in Jannah cannot be called a human being.
B. The test (or the challenge as I prefer to call it) is not at all severe, it is serious but it is not severe. How can we call it severe when God says in the Qur'an that (when we deserve it) He will be willing to forgive all our sins (39:53)? Please note that most of the verses of the Qur'an that are about punishment are in fact referring to the infidels of the time of God's messengers. These were people who arrogantly and while knowing the message of truth, rejected the messengers and their message and decided to go against it. Their punishment is in fact the result of their rejecting what they knew was the truth, only out of arrogance (4:165). Other cases where the punishments are described refer to those who commit major sins like murder and fornication. Even there a true repenting may save the person from punishment.
C. According to the above, it is not accurate and it is misleading to say those who fail the test will be punished. The reality is, only those will be punished who fail the test out of their own arrogance and despite perfectly being able to pass it, while - because of the scale of their failure - they do not deserve any leniency by the Almighty.
2. How can he be RAHMAN in this case? He is so angry with SHIRK that he can put a soul in the extreme torture, how he can be SAMAD?
As explained above, only those who arrogantly disobey their Lord while knowing the wrong nature of their action will be punished (if they are not forgiven by God). Another point to notice is that although it is correct to say that God punishes people, however in fact it is the action of the people that materialises as punishment for them in the Hereafter so those who go to hell have actually fired up the punishment themselves. They are the reason why the fire starts and goes on (2:24). Everything we do in this world has an appearance that we can see and a reality that we can only see in the Hereafter. We do backbiting (Gheebat) in this world and we think that merely as an example it is like eating the flesh of our brother. It is only in the Hereafter that we come to realise and appreciate that 'eating the flesh of our brother' is not merely an example to show how bad backbiting (Gheebat) is, but it is in fact the 'reality' of backbiting (Gheebah). Similarly when we read in the Qur'an that those who possess what belongs to orphans are eating fire, we are not merely reading an example to know how bad possessing the wealth of orphans are, this act is in deed eating fire, one only comes to realise it when one reaches to the Hereafter. In the Hereafter the reality of everything (good and bad deeds) will show itself (50:22) and as the result people who have done bad deeds so extensively and/or seriously that these bad deeds have dominated them and have become part of their personality (2:81) will be punished (unless forgiven by God). In other words, the person who is used to possess the wealth of orphans in this world to the extent that this behaviour has become part of his personality cannot help it but eating fire in the Hereafter because this is what he used to do in the world.
Looking from the above perspective, where people are forgiven, this is because of God's extreme mercy and where they are punished this is merely the result of their own actions and in no way goes against God's mercy.
3. Many ahadith in Bukhari say that Jannah and Jahannum is written on Loh for every soul. A man may do bad deeds for life and at the end he will make repetence and go to Jannah as per his luck and vice versa. What remains for the test then? And where is mercy (Rahma)?
God the Almighty is Himself the creator of the dimension of time. He is not limited in time. Therefore the above Ahadith and the verses of the Qur'an with similar messages are simply implying that no matter what we might do in our challenge of life, God the Almighty, knows what it will be and that where we will end up at the end. This does not mean that it does not matter what we do. It simply means what ever we do, God knows it in advance. In fact, even the use of the phrase "in advance" is inaccurate and only serves the purpose of our own limited understanding, as for God, there is no past or present.