Why did that General guy kill himself in the beginning part?

Firstly, a minor point; Fambrough was a major, not a general.

Fambrough seemed to have mental heath issues. We hear the clanking of bottles when he opens his desk drawer, suggesting alcohol issues. His staff at the office seem used to the major's behavior and mostly ignore it.

HE was a Major

HE was a Major

I'd say he was already dying an alcoholic and severely depressed with little hope left. I don't think speaking to Dunbar had a huge effect on his actions, he had probably thought about it many times before and decided that now would be as good a time as any.

It's in the novel. Essentially, he had slowly gone mad. He didn't want to be there, he was alone, was out of his element, and had no friends.

It's primarily a plot device. No one knows that Dunbar is at the fort because the one officer who knew killed himself. The soldiers that had been at the fort were starving and left on foot to try to get back to town, but were all killed (to my memory) so no one even reported that the fort was unmanned. Thus the long time without resupply or contact.

In the novelization, it makes way more sense but putting it in the movie would slow it way down and it's already a slow movie.

Maybe they'll make a shitty prequel (on streaming) that explains it for people who can't come up with their own explanations.