1 - "His ideology is that gravity shackles a person's soul"
I wouldn't look too much into that term. It's more of a buzz-phrase than anything. His true ideology is that people need to leave Earth so that a)It can recover from the pollution humanity has done to it and b) by completing its migration into space, humanity will further its evolution into space (Newtypes)
2 - "He killed his best friend"
Most people don't know that.
3 - "More to the point, he spent a year (actually only like 3 months) trying to kill Amuro and company. Why are they sucking him off all of a sudden."
Obviously Kai has his misgivings about trusting Char, and Amuro is initially suspicious of him. But Amuro has a very complicated relationship with Char, and I think the others just write it off as nothing personal, that they had just been on the wrong sides of the war
4- " why would anyone give two damns about the fact that Char Aznable was actually the son of Zeon? It doesn't even matter that the Zabis were different from Char's family, because the word is synonymous with them, now."
No, the Deikun family and the Zabi are most definitely not viewed as synonymous by people in UC . Zeon Deikun was literally the most beloved person in the Earth Sphere prior to his assassination. He was a viewed as a guiding light, a true leader and visionary and his philosophies (contolism and ereism, not to mention Newtypism) changed the way people thought. With Char resurfacing and revealing himself to be Casval Deikun, people believed he would take after his father and stand as a man of integrity.