We have seen worse feuds in the team, this is nothing. Ganguly ( & most of the seniors) vs Chappell was way worse. Even Dhoni (while struggling overseas in Tests) going behind Gambhir's back and kicking him out of the team, because the board wanted to make Gambhir the captain, etc was also a worse feud. And these are just the recent ones. Back in the time, Sachin had problems with Azhar, Kapil and Gavaskar clashed, etc.
Anyway, I have nothing against Kumble (he is legend), but he only got the job because the people choosing the coach were his best buddies (Sachin, Ganguly and Laxman). Ravi Shasri (who might again get the job) was doing a lovely job till then, and wanted to continue. Apparently his presentation was also superb, but Kumble got the job.
I believe, that in cricket, the captain is more important than the coach. So, Kohli will get his way here, because he has shown himself to be a capable captain. And this is not new. Dhoni got Fletcher the job, because he wanted a 'yes man', which is what Fletcher was during his time with the team. At least, Shastri has shown to aid the captain and present good ideas (without being a yes man), instead of imposing his views on the captain. And like Kirsten (who was awesome) he was a good man-manager. I hope Shastri or Sanjay Bangar or even Lalchand Rajput get the job, over the others. Or maybe even Moody.
Kohli should be more worried about Rohit Sharma and Ashwin, because there is going to come a time when Kohli might start to fail, and those two blokes might show an interest in captaining the side. Rohit has a done a terrific job with the Mumbai's IPL team. With Ashwin's failings in the limited overs formats, Rohit seems to be the biggest threat. And unlike Rahane (our current vice-captain and a quieter version of Dravid), Rohit is ambitious and vocal.