Are you sure you actually watched the movie?
Cuz some of these questions were answered even more than once throughout the movie.
Maybe it's worth a second watch.

Did Jessup intend to have Santiago killed?
Of course not. Think about it, why would a commander want one of his own men killed? Especially after insisting that he be "trained" to conform to standards?
Why was Santiago silenced? What did he know?
He wasn't "silenced"; his death was unintentional. He was being punished because he betrayed the unit code by going outside of the chain of command to rat on a fellow squad member.
Who fired at the fence line?
Dawson. But the reason he fired is because he thought his "mirror" was engaging him. Santiago was the only eye witness to the event and he's dead. There is no reason to disbelieve Dawson's version of events (he had no motive).
Was the rag poisoned?
No. Santiago died of a heart condition that the doctor failed to diagnose, combined with lack of oxygen by having a rag placed in his throat. It was an accidental death (Dawson & Downey didn't mean to kill him, they were just going to shave his head).
Why did Markinson end his own life? What did he know?
Now this is actually a complicated question, and the answer is ultimately left up to the interpretation of the viewer. Markinson knew "everything", and that was part of the problem. He knew that Jessup wanted him trained, he knew that Kendrick ordered a code red, he knew that code red's were illegal, and he went along with the fake transfer order after Santiago died. He participated in a conspiracy to cover up the death of a marine in his unit. He felt guilty for not having a backbone to stand up to Col Jessup (or report the illegal activity), and allowing an innocent marine to be killed. That's why he went AWOL and eventually contacted Kaffee. He failed in his duty to protect the innocent (probably the reason he joined the marines in the first place), and he ultimately could not live with himself.
What was the truth, and why couldn't we handle it?
There are multiple truths here.
The truth is, the Marine Corps cannot afford to have weak, inept marines (Santiago) "protecting" the country. They either need to be trained to conform to standards, or discharged so that more capable marines can take their place. And allowing them to be transferred to a different unit with more lax standards is passing the buck and not taking responsibility for training marines to be the best. Allowing weak, incompetent, or otherwise unfit people to stay in the corps weakens the force, and diminishes the United States ability to win wars. And the accidental death of an inept marine due to a training mishap (while unfortunate) pales in comparison to allowing such a marine to have other people's lives put in his hands, and them being killed due to his incompetence. The reason we can't handle the truth is because that is a hard pill to swallow for the average person (especially if they never served). Santiago seemed like a nice kid. An innocent guy who didn't deserve to die senselessly. But he was not a good marine, and really never should have been in the corps in the first place (given his medical history). That is the truth that Col Jessup wants you to hear to distract you from the OTHER truth.
The other truth is that Col Jessup is a liar, a coward, and a disgrace to the Marine Corps. He is motivated only by power, position, authority, and promotion, and because of his ambition he lost sight of what was really important to a marineHONOR! He acted dishonorably by disobeying the standing order to actively discourage "code red" discipline. He instead promoted it! Dues to his poor leadership, he placed one of his own men in danger by subjecting him to an unregulated, illegal training style. And he acted even more dishonorably by lying about the circumstances regarding Santiago's death. He pretended that he was being transferred to another unit to cover up the fact that his death was the result of code red style "training". And he hung Dawson and Downey out to dry, hoping they would take the fall for him. He put himself and his career ahead of his unit, and because of that one of his marines died, his second in command committed suicide, and two of his enlisted men were wrongfully accused of murder.
This artist:

It's that kind of movie, there are things we just don't know and it's why I love it because it makes YOU decide based on your own interpretation of the characters and the evidence. The only thing we know for a fact is that Jessup ordered the code red and had to be badgered into admitting it.
However what we don't know:
Who decided what form the code red should take? Jessup, Kendrick or Dawson and Downey?
Was the rag poisoned with an undetectable toxin like the Doc said?
Doctors and coroners change their diagnoses all the time, was the Doc actually coerced?
etc etc etc
Hey! You're not old enough to drink! Now go and die for your country!!!

Paul_Ke: That makes sense. Thanks for the explanation.
Fowler's knots? Did you say fowler's knots?

Markinson killed himself because of his sense of honor. (This doesn't make sense, though. He would have demonstrated more honor by testifying at the court martial.)
Markinson killed himself because he felt ashamed to blow the whistle and testify against his fellow marines, he felt like like he was betraying them. He felt trapped, not willing to testify against them but not willing to do nothing either.

Markinson killed himself because of his sense of honor.
(This doesn't make sense, though. He would have demonstrated more honor by testifying at the court martial.)
It's funny you should say that. I started a thread making that very point a while back but the OP actually made a point that made me think.
Why did Markinson end his own life?
What did he know
?
Is it possible Markinson was implicit (unwillingly) in the plot and knowing Jessup's machinations couldn't afford to take the stand and risk being cross examined by Captain Ross?
Even if he wasn't implicit Jack could have asked him why didn't he report Jessup and Kendrick to the authorities for ordering the code red or for making the flight disappear. In fact when you think about it Markinson was probably a less credible witness than Downey.
Hey! You're not old enough to drink! Now go and die for your country!!!

This message has been deleted.

I would answer all lf you questions but, You Cant Handle The Truth!!

Jessup intended to punish Santiago, not kill him, to motivate him to become a good Marine.
The rag wasn't poisoned.
Markinson killed himself because of his sense of honor. (This doesn't make sense, though. He would have demonstrated more honor by testifying at the court martial.)
The truth is that you need Jessup on that wall, because it saves lives. (I don't know whose lives it saves; you'll have to ask Jessup when he gets out of prison.)
Fowler's knots? Did you say fowler's knots?