I'm a few years out of the game myself, and from a different intel branch, but I'll take a stab at answering your questions:
So what you're saying is that an NSA agent must prove that someone is a foreigner before collecting data on them? To whom do they have to prove it? What are the consequences if they fail to do so?
To my understanding the check for whether a subject is a U.S. Person should happen before any intrusions on their privacy occur. In practice, though, the investigating team can do pretty much whatever they want provided that they don't care to press criminal charges. If a court case ever were to occur, the investigators would be asked by the judge to show evidence of their due diligence. There are many other avenues for neutralizing intelligence threats that don't rely on judicial action; neither stripping government employees of security clearance nor deportation of non-citizens requires a judge or compliance with constitutional protections. The main consequence of violating a subject's constitutional rights is that any court case to prosecute will have illegally obtained evidence thrown out and will likely fail.
What actual consequences would an NSA agent face if they did ignore that fact? How would it be discovered? How often has this happened?
So far the only actual safeguard you've offered is "trust us". Can you at least try to understand that that's not good enough?
To the NSA? I don't know. Maybe nothing. I don't know how it would be discovered, especially if the actions taken in response don't involve courts. And there's no way to know (from the outside) how often it happens. For what it's worth, I agree that "trust us" is not good enough, and that it would be better to have a judge sign off on anything questionable as a matter of policy and standard procedure, even if it were after the fact. At least there would be someone capable of reminding them when what they're doing is unconstitutional.
So the president is complicit in the unconstitutional wiretapping of US citizens and that's supposed to make us feel better?
Nope, I don't feel any better about that at all.
Also, as a technical matter, how does one capture the packets of foreigners without also capturing the packets of citizens? At the very least, doesn't the NSA have to store and analyze the packet to determine whether it belongs to a US citizen or not? At that point, hasn't the law prohibiting collecting, storing, and analyzing the communications of US citizens already been broken?
I don't know the answer to that, either. "Advanced filtering" sounds a lot like "trust us". Analysis in RAM without storage to disk I think would be OK, but it doesn't sound like they're doing it like that. I think they're on the wrong side of the line there, but it's the side of the line that lets less data slip away so I can understand how they got there (even if I don't agree with it).
The Director of National Intelligence just recently admitted that some NSA activities had violated the Constitution at least once.
By whom, and what consequences can we expect this criminal to suffer?
See above; probably nothing. Just like cops running a bad investigation don't get fired when they botch it on constitutional grounds, intel agents don't go to jail for violating U.S. citizens' rights.