Can individuals spend money acquired in settlements that include an NDA? is the party that receives the payment allowed to actually use it?
Yes. Payment of a substantial sum is oftentimes the main or only incentive for the injured party to bind himself in an NDA. A prohibition to spend that money would de facto nullify the benefit without which the party would have declined the NDA.
Sandman isn't otherwise wealthy, if he suddenly buys a Lamborghini or a private jet, that money must have come from CNN. So doesn't this violate his NDA?
No, typically it does not.
Depending on (1) who the parties are, and (2) the hitherto divulged details of the controversy, the public might infer anyway --and with reasonable certainty-- who must have paid a substantial sum to settle the dispute. Thus, plaintiff Sandmann's [hypothetical] purchase of luxurious items would not really provide the public with new information.
Similarly, if instead of buying luxurious items the indemnified person spends that money on rather necessary expenses (be it medical, property & casualty, etc.) ensuing from the injury, a prohibition to spend the money would lead to another absurdity: That of perpetuating or worsening the harm inflicted to the plaintiff who couldn't otherwise afford medical treatment, new living arrangements, or repairs/replacement of property, accordingly.
If the payer thinks it is important enough, perhaps the NDA could include terms requiring the payee to use some extent of discretion as to the pace of spending the money. However, such need on the payer's part seems rare, and the injured person is likelier to decline the NDA altogether if the clauses for management of the compensation are unreasonable.
What counts as disclosure with regards to Non-Disclosure Agreements?
Any revelation, by the parties, that contravenes the terms and purpose of the NDA in that it would give non-parties information which cannot be ascertained from prior and/or independent publications and filings. That is otherwise hard to answer because it totally depends on the terms --and boilerplate-- of each NDA.