However, if a newspaper prints a story saying senator X said Y, and that claim was false, then the newspaper can be sued, regardless of where they got that information or which employee wrote the story.

Given your analogy, the question is, is Facebook more like a car manufacturer or a newspaper?

If a newspaper runs an advertisement that turns out to be false (say, a fake claim about a thing for sale), is it the newspaper or the advertiser who's at fault?

The advertiser. But if someone sues the advertiser and it turns out that the person listed by the newspaper as the advertiser isn't actually the advertiser, it's the newspaper's fault for not verifying their customers.

So it is the newspaper's fault that their customer lied to them?

There are certain industries that have requirements for vetting customers, and if those requirements are not met then responsibility falls on the company and not the customer.

If a car manufacturer distributes a car that did not go through any inspections and it turns out the brake lines were hooked up incorrectly and consequently causes damage or hurts the customer then it's neither the customer or the dealership who are at fault, but the manufacturer for failing to properly inspect the product that they are making.

In the same sense, if a bar sells beer to someone without asking for their ID and they turn out to be underage, or a gun store sells a weapon to a convicted felon without performing a background check, liability falls on the company and not the customer.

I am not sure if this should apply to newspapers in respect to their advertising, but there are examples where this line of thinking does apply and this might be one of them.

That is a wrong analogy. FB does recommendations prominently luring in you to things that you didn't knew existed. It also filters and sorts to decide for you what information you get to see.

The correct analogy is having a secretory that reads all your emails and only selectively lets you know what you got as well as inserting recommendations based on her belief system. So if this secretory tells you that your brother only sends you bad emails and here's the great model of gun you should buy if you are going to meet him - then your secretory is not merely a tool to manage your emails.

Horrible analogy. If Ford had to approve every ignition/engine start request and they had a breathalyzer installed in every vehicle that would send the blood alcohol level to Ford's API and required a signed response before the engine would start, maybe you're getting close...

Facebook put in a "vetting" process and requires everyone to go through it to publish ads. On the other hand, anyone with enough cash can buy a Ford (even second hand from a 3rd party), get drunk, and drive through a few neighborhoods causing death and destruction.

Okay, maybe a better analogy would be with seatbelts. It doesn't seem like it would a huge engineering feat to prevent a car from starting if the driver isn't wearing a seatbelt. If someone dies in a car crash because they weren't wearing a seatbelt, is the car manufacturer at fault?

One thing I'd point out is that a drunk driver, dangerous as he is for individuals in his path, can't really affect anything beyond his immediate vicinity. Facebook has national and world-wide influence.

Drunk people do not install presidents.

Neither do ads on the internet. Donald Trump's presidency did not mesh well with a large population of the US's perception of reality. It is much easier to say "the boogey man caused this to happen" vs "my understanding of the world around me was inaccurate."

I'm not saying that there weren't campaigns that intentionally mislead people or that those campaigns aren't a bad thing. I do think that blaming the Trump Presidency on the Russians is overly simplistic and is an easy way to ignore a real divide in our country.