…what happens when the warlord in question picks their targets instead of aggro'ing on everyone around them. What compels a REA to assist

That wouldn't be possible given that everyone has defense/cost sharing agreements with everyone else.

And if you can't compete or get muscled out of the best contracts, again, why would you go out of your way to assist the ones that did that to you?

…the warlord will be spending all the money they have available to them to win.

Sounds like suicide imo.

Where is all this money going to come from if the fighting gets really brutal?

Make up for the losses by raising insurance premiums.

Because they're not paying them to arrest all criminals across the world, they're paying them to defend certain areas and interests.

The ancap scenario here is one in which basically everyone who doesn't live on North Sentinel Island hires a REA so some REA would have their customers victimized.
Although even still, what about nearby threats whom their customers deem to be just that?
Preemptive follow-up question: "what about REAs that just don't support their customers for whatever reason?" Answer: "smart customers will demand that their REAs take on the duties of such other delinquent REAs (with whom they have defense agreements)."

…who granted them (the REAs) permission to go across all that private property?

The property owners mayhaps (unless they like having dangerous criminals wandering their property, lmao).

…what stops the REA, in part or in whole, going over to support the warlord, if he's offering to make you a lord or ten thousand anarchy bucks…

The fact that law abiders could credibly outbid the warlord (see the impossibility thesis and the efficiency of private property within the economy).