You can already sue them. For example, if Fox runs a segment claiming that you (gulli1010) are a murderer, you can sue them for slander (with or without this executive order).
If someone tweets that you are a murderer, you can't sue Twitter (or rather, if you do, Twitter will be protected by Section 230).
However, suppose Twitter announces the following policy: "Any tweet which calls someone a murderer, shall be deleted, unless the alleged murderer is gulli1010". Should you be allowed to sue Twitter then? That's a more nuanced question. And what if the above policy isn't an official policy but is a de facto policy?
People should look into the Seth Rich lawsuit and Ed Butowsky. Trump forgets his crowd is the one who constantly panders conspiracy theories like PizzaGate and QAnon. This will effectively take them down entirely.
No one side has a monopoly on conspiracy theories. The Russia collusion was something spun for years by the networks where people under oath said one thing privately and another publicly.
Yes, but at the cost of the entire platform. I’d doubt that Twitter could sustain being endlessly sued over user posts. They’d die the death of a thousand cuts. And a lot of conservatives would be perfectly happy with that.