Important question. Here's some ideas I've come up with or stolen from Stef's podcasts.
1. How would a DRO that is lying about weapons purchases and hiding
weapons from 3rd party inspectors be stopped from taking over violently?
Weapons that are used for offense are very expensive and would lead to an increase in the rates of those who are customers of the DRO. Not only would this discourage those customers from continuing to support the DRO's, but such an increase in prices would also raise enormous red flags.
2. Couldn't a DRO just lie about the "10 million dollar deal" just
long enough to become organized and powerful enough to take over?
DRO's that suddenly spend 10 million dollars would probably need to raise the rates of their clients. In order to keep customers from leaving, DRO's would have to strongly convince their customers that this raise in cost is justified. Customers might ask that a third party inspect the purchases made by the DRO.
3. What if a DRO grows quite a bit and then a new guy comes in and starts murdering people and tries to start a new state?
Do you mean a new head of the DRO company? I do not think a DRO as a whole would mindlessly march into battle at the command of a new leader who has gone insane. However, even if this did occur, this DRO would rapidly lose its funding and other DRO's would be able to charge slightly higher rates in order to fund an effective solution to this out-of-control DRO.
4. How would competing, individual DRO's fight against a large-scale
attack such as nuclear/biological weapons or a large scale invasion
from a well-armed nation?
This is one of the toughest questions for me. These DRO's might have an incentive to work together under this circumstance given that they would otherwise lose their entire businesses.
5. How many people would actually have fears or care enough to be
constantly watching the DRO for any suspicious activity (I'll admit
it's a cynical question to begin with, but I didn't have an answer)?
IMHO, no political system or lack of a political system would work if a certain percentage of the population wasn't willing to do a bare minimum. Asking how a DRO system could work if people weren't willing to watch over the DRO would be the same as saying "how would a credit rating agency work if lenders never checked credit ratings." This is a very difficult question for me, though others might have better answers.
6. Wouldn't a DRO be unstoppable if they found just one corrupt bank willing to not seize the DRO's assets?
That would badly harm the reputation of this DRO.
7. Wouldn't DRO's just turn in to large gangs, fighting each other
for power? (I said that people wouldn't fund or support them if they
did this, but they then noted that by this time they could just start
subjugating the people under their respective contracts)
If they tried to subjugate them through contracts, there would need to be a clause in the contract that makes the contract void if the DRO's violate the non-aggression principal.
These are just my initial responses to the questions.