How to submit required forms to customs when leaving USA?

You'll need to visit the Customs office yourself, which is often in some obscure corner of the airport outside the secure area, possibly in a different terminal from the one you're in (allow plenty of time for this). It is your responsibility to go seek this out in advance, and they've seized cash from travelers who didn't do so.

You can use CBP's Locate a Port of Entry website for contact information. For example, at JFK, their main office is off in building #77, somewhere in the cargo area. But more importantly, that page lists phone numbers. Give them a call, tell them you need to make a currency declaration on your way out, and ask where to go. They'll know the local procedures.

Or you should avoid carrying more than $10,000 in cash or other declarable goods. (By which I mean don't carry large amounts of cash at all. Carrying amounts just below the threshold to evade reporting requirements is illegal.) A bank transfer is generally the better option for most travelers.

Should the passenger simply inform the passport control officer that they need a customs officer? Or maybe passport control officer doubles as customs officer at the same time? Or should the passenger simply make an advance visit to some Customs Office in the airport?

You won't speak to a "passport control officer" on most departures from the USA. Unlike most countries the US does not have passport control for departing passengers.

Also, if an international flight out of US begins with a domestic leg, what would be the proper location for dealing with the customs? Can it be done at the first airport?

You will need to seek out a Customs and Border Protection officer yourself. Your first domestic airport may or may not be an international airport with a CBP presence, so you may need to do this at your final stop in the US.

Here is how it turned out in my specific case:

1. I contacted Customs Office at SFO by phone approximately one week in advance and asked them this specific question. I explicitly explained to them that I have a layover at JFK. Nevertheless, they told me that I can submit my customs declaration at SFO.

2. Just in case, I contacted Customs Office at SFO by phone again two days before departure: to reconfirm and ask about the details. Again, I specifically pointed out that my last stop in USA is JFK, not SFO. They reiterated their original answer (a different person this time). Since my plane was departing rather early in the morning, they told me to come to the International Arrivals area at SFO and call a specific phone number to have a customs official come over and escort me to the office.

3. Just as they instructed me, on the day of my flight I arrived early, came to the International Arrivals area of SFO and started calling the number they gave me. Originally there was no response. I had to call it repeatedly for quite a while. Time was running out. Eventually someone picked up the phone (a third person, judging by the voice). This time the person on the phone sounded quite surprised about the fact that I wanted to submit my declaration at SFO. They told me that in this situation I had to submit my declaration at JFK, not at SFO.

4. Luckily, I was able to find the Customs Office at JFK without any problems and submit my declaration there.