If you beat the main quest. Does it matter if you kill the king and his throne room guards? I really want to get that ring that he has and the Grave Digger Sword, that the guard captain has. Also, another unrelated question, what is Alamexia's soul used for if you get it in Azuras Star?

Not sure about the king; I've never bothered to kill him.

Almalexia's soul is great for cast on use or cast on strike enchantments. Don't waste it on a constant effect that you can get from a normal golden saint.

Killing Helseth will result in you getting the warning that you have broken the prophecy, but it won't actually ruin the Morrowind main quest.
If you kill him before completing the quest where you fight Karrod, then the Tribunal main quest will be broken. But after fighting Karrod, you can kill Helseth without breaking it.

Most people in the world will talk about his death, but no one seems to be upset at you.

Killing Helseth after fighting Karrod may not break the TB MQ but you will miss out on a free set of Royal Guard armor. Granted the armor isn't great and most players will have better stuff by this time but there are those that like to collect full sets of armor.

I suggest taunting Helseth into attacking first and make sure he actually lands a blow before you hit him. Otherwise you will find yourself fighting everyone in the room.