What are the rules or restrictions regarding cooking in a charcoal grill on Yom Tov (besides needing to use an existing flame to light it)? Can I add lighter fluid to the charcoal before lighting? Can I light the charcoal in a ’chimney-starter’ (where burning newspaper is used to start coals)? Can I move the coals around?

You can use a charcoal grill on Yom Tov to cook for that day only as long as you use existing fire to light it and you don't extinguish the fire when you are finished. The burning coals can be moved only of it is necessary and essential for the actual grilling. You can add liquid fluid before. (Shulchan Aruch OC 502, 507)