To your first question, an emphatic yes. That's the point of labour laws, employers rarely fire people for fun, they do it for economic reasons.
I expected your second and third questions, and I agree with the premise to some extent. If algorithms promote or demote some speech they affect free speech. If you accept that, free speech does not exist online and never can by definition.
It's a spectrum however and it's hard to reason about because recommendation algorithms are complicated, have many parameters and are a black box to consumers.
If a company promotes posts that benefit it financially, that's very far from free speech, if a company promotes posts based on a random generator it's very close to free speech but probably useless to users.
I don't have a bite-sized answer to what must be done, it's a project that must be started and tested in the real world. I'd certainly prefer the government to be in charge of what gets implemented, it's a matter of public good that should not be left to private corporations. My naive, off-the-top of my head idea would be a range of public, open source algorithms cautioned by the government that prevent companies from exploiting their users. There are other softer ideas depending on your beliefs wrt the government, for example you could let all companies use any algorithms but force them to publish them.
To me social media is equivalent to tobacco, sugar and gambling, it's highly toxic, exploits our baser instincts, and cannot be solved by an individualist perspective but social media is a fairly recent phenomenon so I think it'll take a few decades before we grasp the full severity of the problem.
W.r.t the press, I think the problem is apparent but I'm hesitant to answer because I'm not knowledgeable enough about the domain. Often technological solutions get discussed here, like micropayments & cie, which don't seem to be doing well. I think there is something to be said for more diversity in the industry so it's more representative of the people they're supposed to inform, but positive discrimination is a tricky subject. There are also lot of former government employees working for big media companies so we're not starting from a blank state wrt the influence of the government, passing laws could rebalance things but I honestly don't know what form they'd take. I know addressing perverse incentives doesn't have to lead to a curtailing of free speech.