Aka shade tree mechanics. Like me. Can do many basic things but sometimes need a scan run to see what a problem is so I don't have to replace 4 different parts trying to guess what is wrong. Haven't had a car of mine in a repair shop for 15 years and all 3 of them run great.
yes sounds like the head gasket but also could be the intaake manifold gasket and my advise is to buy a new car
Don't complain. It was bought "AS IS WHERE IS" and I'm sure that a NO WARRANTY release was signed at the deal.
There is no way to know when a head gasket is going to fail. Your sister may also have caused the bulk of the problem. She drove it for 4 hours, probably pretty fast (I've made that trip many times, everyone goes fast!). Engines don't just "shut off due to overheating" without warning. There had to have been a gauge on the dash that she paid no attention to.
You said, "She later noticed a crack in the radiator" and also said "She poured some water in the radiator." Cold water on hot parts can cause trouble! You also say "I also know that radiators do not just go randomly" well yes they can! I'm sure that the overheating during the trip caused excess pressure build up in the radiator. The tanks are plastic, they can't handle too much.
Now I am not saying that this was her fault, but I am saying that had she noticed the overheat and immediately had it checked, the cause could have been a minor trouble and the major trouble may could have been avoided.
Again. Don't complain ( as in whine, demand and act like the car lot owes you something, because legally they do not. You can however inform them of the problem and ask if they can help with repair costs, or another vehicle. Being so far away, I doubt they will but you can try.
One other thing here, You also said; "I know that dealers need to do some type of inspection on there cars before they are able to sell them" That isn't really true. There are some state inspections that must be done in some states, not sure about Nevada. There were no state inspections when I lived there. No laws require any inspection of longevity though. As long as the brakes, lights and safety stuff is good, it can go!
Your sister ought to complain to the independent mechanic who did the pre purchase inspection for her! ( I am betting that didn't happen huh?).
Sounds like a blown head gasket.