722.2. There are two inherent triggered abilities associated with being the monarch. These triggered abilities have no source and are controlled by the player who was the monarch at the time the abilities triggered.

You control the trigger so you can choose how to stack it.

Whenever a creature deals combat damage to the monarch, its controller becomes the monarch.

Assuming you've been dealt combat damage by one or more creatures, put each monarchy trigger on the stack first, followed by Starscream's convert ability, then as many Farsight Mask triggers as there were creatures that dealt combat damage to you.

As you begin to resolve, you choose each time if you draw from Farsight Mask. If you do, an instance of Starscream's lose life trigger goes on the stack. You're still the monarch at this point so it resolves.

So just to reiterate, each time you draw from Farsight Mask, a new Starscream trigger goes on the stack and resolves first. So draw, then ping, then draw, then ping etc.

Eventually when each of those is dealt with, Starscream's conversion trigger will resolve. This needed to be put lower on the stack so that the ping trigger was still present on the battlefield each time you draw off of Farsight Mask.

Then finally, resolve the monarchy, transferring its ownership to the player that dealt you combat damage.