State vs. federal
Given that the plan is state-dependent, yet schools are located nationally,
I don't believe that this is a correct statement of the plan status. A 529 plan is established as an entity by federal law and gives federal tax benefits. A 529 plan may also have state tax benefits. As such, the rules for the plan are set by the federal government for the federal tax benefits. You have to comply with those federal rules to get the federal benefits.
They cannot create a state 529 plan that gets federal benefits but does not have to comply with the federal rules.
They could create a state plan that does not get federal benefits and does not have to comply with federal rules. But such a plan would not be a 529 plan in the normal sense, as 529 relates to a section of federal law specifying such plans. Such a state plan would have a different name. It's also worth noting that this is unlikely, as the federal tax benefits are generally larger than anything that the state could offer. The highest income tax at the state level is in the 10-15% range.
A state may have additional requirements to get the state tax benefits, but they can't remove requirements to get the federal benefits. This is further complicated by the fact that the benefits come long before the restrictions. So what would happen if you didn't comply with the federal requirements is that you'd be forced to pay the federal government a penalty and taxes. The state may not bother charging you taxes in that situation, but the state decision won't affect you federally.
I doubt that the location of the school matters. The federal tax benefits are enough of a nexus for regulation in this area. They don't need the commerce to be interstate. But that's really more a question for Law.SE.
Anyway, your state may not care if you use your funds for this purpose. They may not revoke any state benefits that you received. That still won't keep the federal government from revoking the federal benefits, which comes with a penalty. Since many states have much smaller income taxes than the federal government (up to 39.6%), the federal benefits are going to be what's important to most people.
In your particular case, Nevada is one of the states without an income tax. So you received no tax benefits at the state level, only federal benefits. You must comply with federal rules for those benefits.
The Bankrate.com columnist Dr. Don addressed college application fees:
Some schools specifically indicate application fees are NOT considered a qualified education expense.
Even if that argument were allowed by the IRS, it could only apply to the one school you actually attend.
I believe it’s not worth the trouble or tax risk to pursue this path.
Some schools explicitly tell you that application fees are not a qualified expense. Presumably they do this because other applicants have tried to use 529-funds and had to pay taxes.
The argument that they should be is such that you could only claim them for the one school that you actually attend. You'd have to pay taxes on all the checks to other schools. Or somehow only use the 529 account for the one school that you actually attend. Problematic in that you submit the application in order to find out if you can attend that school.
And of course remember that the IRS doesn't have to agree with this interpretation. They might fight it in court and win. So even if you successfully picked the single correct school and only used the 529 for that application fee, it still might not count as "expenses related to enrollment or attendance at an eligible educational institution."
Other forums are more definite and just say no.
The IRS produces Publication 970 if you want to read it directly.
TL;DR: No, it doesn't seem that application fees are qualified expenses for 529 plans.