I will preface this post by saying I have a very realistic approach to the Bhagvad Gita, I do not eulogize it as some holy book. The Bhagvad Gita was written by some anonymous author who added the Gita to the Mahabharata, this most likely happened during the philosophical period when the battles were raging between the rival schools like Vedanta, Samkhya-Yoga and Buddhism, and the Gita was the earliest attempt to try and reconcile these contradictory viewpoints. This is why the Gita is quite a condradictory text, saying one thing in an earlier chapter and later saying the opposite thing. It was also the first attempt to give a human and personal face to the impersonal Brahman of the Upanishads, giving it the face of Vishnu, suggesting that it was probably an author within the Vaishnavist religion and hence the obvious Bhakti bias in the text.
Now to answer some of your points:
It is also written that krishna made the cast system so people would know their place , and they would get the birth by their qualities.
No, Krishna says that he(speaking as god) made society according to the gunas. Traditional Hindu society was made up of the four classes: brahmin, kshatriya, vaishyas and shudras. This social system is as old as Vedic times. So we see here a divine explanation is given in the Gita to justify the system.
Also it says only the hare krishna mantra will enlighten people in the age of kali. This is what the hare krishna movement also says .
The Gita does not say this, this comes from a far later modern Vaishnavist movement known as Gaudiya Vaishnavism. The ISKON belong to Gaudiya Vaishnavvism. The foundations for Gaudiya Vaishnavism comes from the Puranic period where there is a huge shift to bhakti, idol and temple worship. It is here these kinds of lazy beliefs are propogated that merely saying the name of god is enough to bring ones liberation, and it is justified by claiming it is deemed the best method to liberation in the Kali yuga.
Also the aryans conquered rest of india who knows if they didnt put this in the book that people is born into the cast by their qualities ,to control the others, like other religions do. This is not impossible.
Highly contentious. Aryan invasion theory is today hugely controversial. It should not be stated as fact.
And bathing in ganga to remove your sins is one of them
This is not in the Gita. This is just Puranic stupidity added later.