SPOILER! Don't open unless seen.

That's where I thought the plot was going also, but the movie never really explained Kyle's death.

Yes he did on purpose.

My take after watching the movie was this:

Max was running ahead when the explosion went off. Kyle ran forward for Max not knowing if he survived the suicide bomber. Then the bullets started flying from both sides. When Kyle ran forward towards Max, Tyler should have either pulled him back or started shooting to cover Kyle. However, he didn't do either and jumped and hid behind the wall because things would be easier for him if Kyle was dead. The other men in their company had to yell for Tyler to get back in the fight. By the time he stepped out from behind the wall, the fight was already over. This is also why he lied to the Dad regarding Max being responsible for Kyle's death because really Tyler was responsible for not pulling him back or covering him in the fire fight. It's plausible though that he shot him before moving behind the wall to take cover. The movie never really fully explains it.

After the dust cleared you see Max jumping and very agitated. Kyle grabs the dog and holds on the him to calm him down. You then see Tyler has his gun aimed at Kyle and the dog but lowers it when all the other Marines group around them. Then the scene ends. I was under the impression that Kyle was fine and something happened to him off screen. But I could be wrong.

We saw Kyle threatening to expose Tyler's weapons theft. Tyler had to shoot him. We don't know how long after the explosion this took place. Maybe in the heat of a bitter argument, maybe Tyler's partner shot him, if he had one? He likely just shot him, because he felt he had to, to avoid prison.