On your first question, anyone can claim to teach true Christianity. The only way to judge whether such a claim is true or not is to compare what said person is actually teaching against what Jesus himself taught. I know who Ramakrishna is, but am not aware of Vivekananda, but here is my comment. Both of these gentlemen are Hindu and believe in reincarnation, in pantheism, in an uncreated eternal universe and that God is not a personal God. None of these things are true according to the things that Jesus said. Jesus believed that the universe was created and the he preexisted the universe, as is proved by John 1:1. Obviously, Ramakrishna does not agree with this because Hinduism requires that the universe was not created. Ramakrishna does not agree that Jesus is God, but Jesus said that he is God, as proved by John 8:59-59 in which Jesus claims to be a self-existent Creator/God. Ramakrishna has a theology which is pantheistic, while Jesus was a theist. The idea that a Jewish Rabbi would be a pantheist is quite obviously not correct, historically. But that Jesus did not agree with pantheism is also proved by the gospels.
The only way that a Hindu can claim that Jesus taught pantheism is to also claim that the gospels are not true or have been corrupted on a massive scale. Of course, this is what Hindus always do. There is a big problem with this thesis. There is NO EVIDENCE to support this claim. I am not exaggerating. This is a fact. There is literally zero evidence that Jesus or Paul or Peter or anyone important in the Jewish or Christian scripture was ever a pantheist. There is also literally no evidence that a later person changed the Scripture to remove the pantheism of Jesus. I challenge Ramakrishna to provide any evidence to the contrary. Again, since they have no evidence, they resort to claiming that the Bible was corrupted. But again, they have no evidence for this claim. Mark was written somewhere around 30 years after the ministry of Jesus (not 40 to 100 as this gentleman falsely claims). Matthew and Luke were both written within 30-35 years of Jesus’ ministry as well. These Hindu gentlemen are right that John was written later–more like the 80s AD, but even that is only about 40-50 years later, certainly not 100 years later, as he falsely claims. We even have a manuscript of John in the Rylands library in Manchester from 125 AD!
But back to Mark. This account of the ministry and teaching of Jesus was written within 30 years of his life, and it was based on oral tradition and eye-witness accounts which went right back to the very beginning of the Church. As far as I know, all scholars agree that there was an oral tradition about Jesus from the beginning that Mark was aware of. The people who wrote these things down did so when the original people who knew Jesus were still alive and part of the church. The idea that his teachings were lost is ludicrous. The idea that Mark or Matthew could have created a fake Jesus is nonsensical. I have been a Christian for 42 years and I certainly can remember what I was taught when I became a Christian. People’s memory of events they witnessed and words they listened to does not disappear in 30 years. This proposal is really quite obviously absurd.
Also, if we cannot trust the people who actually knew Jesus, like Matthew and Mark and John, then why should we trust a pantheist non-Christian two thousand years later? This is patently absurd! The idea that a Hindu who rejects all the words of Jesus, spoken by him and written down by those who heard him can speak for Jesus 2000 years later boggles the mind. The fact that anyone would actually believe this shows that they are struggling with being gullible.
On the third point, this makes no logical sense. Let us suppose that John was written after 90 AD. Most scholars prefer a date closer to 80 AD, but let us accept this proposal for the sake of argument. Now this Hindu gentleman says that because John was written after 90 AD, that means that Jesus never said “no one comes to the father except through me.” How does a proposal that it was written 60 years later prove that he did not say this? This man can argue that because John was written 60 years later we cannot be quite as sure that Jesus said he was the only way to God, but instead this guy says that it is proof that he did not say this. Where is the logic here? Besides, Luke wrote in the 60s AD that Peter said the same thing (that Jesus is the only way to God) in Acts 4:12. Do these Hindu teachers propose that Acts was written after 90 AD? I do not think so. Are they calling Peter a liar or Luke a liar? I am sorry, but point #3 above makes literally absolutely no sense and should be completely ignored.
On the fourth point, that is the opinion of some Hindus, and I am sure that they are entitled to their opinion. Hinduism does not have a personal loving God. The “god” of Hinduism is an impersonal. Hinduism has an impersonal universal soul which is not capable of loving individuals in the way that the God of Christianity loves people. But the God of Christianity is a loving God who is also a holy God and a God of justice. Christianity includes the idea of ultimate justice. Hinduism does not. Either this is true or it is not true. Let each person decide for themselves if they believe in the God of the Bible. I believe in the God of the Bible because Jesus fulfilled prophecy, did amazing miracles and was raised from the dead. Jesus was God-in-the-flesh and he is the only way to God. If this is true, then even if I did not like the existence of hell, that does not mean that hell does not exist. In this universe there is danger, death and disease. If I do not like this, they are still real. God is a God of justice. If I do not like that, it still is true. These Hindu teachers may not like the truth about God, and they may say whatever disrespectful things they like against the true God of the universe. That is up to them. The God I worship is a God of love, but also of holiness, truth, power and justice. He has proved himself through obvious and undeniable evidence which Hindu teachers do not have. That is my response to point #4.