When Bhagwan Parshuram sees Karna's tolerance, he says that Karna can't be a Brahamana, he must be a Kshatriya.
1 What is the relation between a person capability and his caste?
2 If there had been a relation, then what was wrong in policy that only Kshatriya allowed to learn archery , as they have more strength?
Kshatriyas were warriors so it was believed that they are the strongest and have most pain tolerance among the four prevalent castes at that time.
Bhagavad Gita 18.43 Prowess, splendor, firmness, dexterity and also not fleeing from battle, generosity and lordliness are the duties of the Kshatriyas, born of (their own) nature.
Absolutely there was no rule that who could learn archery and who could not. But Parshuram had taken oath to not teach Kshatriyas after an incidence where he killed all the Adharmi Kshatriyas. Therefore he taught Karna till he believed that he was a Brahmin.
It's true that Parshuram had taught Bhishma. The reason is because as soon as he was born her mother Ganga took him away from his father. So although he was Kshatriya by blood he was just a son of goddess Ganga when she took him Parshuram for his education. So Parshuram had accepted him as his student; in simple words, he was not a Adharmi Kshatriya.
What is the relation between a person capability and his caste?
There is some relation. Usually a Brahmin is associated with knowledge and learning and a Kshatriya with valor and virility. So, a Kshatriya will have naturally more power to tolerate pain as they will get often wounded during wars.
Manu Smriti 2.155. The seniority of Brahmanas is from (sacred) knowledge, that of Kshatriyas from valour, that of Vaisyas from wealth in grain (and other goods), but that of Sudras alone from age.
So, that's how Parashurama had guessed although I don't think that the method is full proof (i.e it may not work all the time).
Regarding your other query, Kshatriyas obviously have every right to get trained in warfare, as participating in wars is one of their prime duties, but Parashurama had vowed to train only Brahmins and not the Kshatriyas. Parashurama's enmity with the Kshatriyas is quite well known.
And another reason why Parasharuma was terribly upset with Karna is that Karna had lied to his Guru about his caste. Lying to the Guru is of course a sin and apart from that pretending to be someone from a higher caste is also a serious sin as the following verse says:
Manu Smriti 11.55. Falsely attributing to oneself high birth, giving information to the king (regarding a crime), and falsely accusing one’s teacher, (are offences) equal to slaying a Brahmana.
Karna, in order to acquire the training, pretended to be a Brahmin in front of his Guru fully knowing that he is not one.
