This doesn't seem to be true. This site [1] lists many agencies (based in the US, UK and other countries) that will sell you a passenger ticket for a cargo ship.
Availability is probably limited though, as each ship can only take a few passangers: "Cargo ships can carry a maximum of 12 passengers although most vessels only have accommodation for 4 to 6 passengers" [2].
You're right. Things seem to have opened up a bit. For a while, security was a big concern. CMA CGM will take passengers on some freighters. Maersk, the biggest line, won't.
That is accurate; there are very very few non-crew berths available on cargo ships nowadays. I know because I've researched alternative means of travelling to see the 2017 Solar eclipse.
They're offered as 'cruises' these days; you'd be looking at rates of ~$100-150/day, and they do single-day port calls wherever the ships' routes make a call. It looks like you do have to book fairly far in advance, though.