The House of Representatives controls the money for the FBI. The Senate, as you might know does not.
The FBI should be anticipating a haircut.
https://thehill.com/opinion/judiciary/3770060-with-new-twitt...
Jack Dorsey can certainly be prosecuted for perjury.
https://www.foxnews.com/media/twitter-files-flashback-jack-d...
Unfortunately, I think neither of the scenarios you described are likely. The House proposes funding bills, it's true, but those bills don't pass without Senate and presidential sign-off. Similarly, if somebody lies to congress, Congress can refer them to the justice department, but it's the executive branch that would execute the prosecution. Even if one could pin Dorsey on a specific lie (and it would be really easy for him to claim that they don't silence conservative voices, just their perception of unfactual information), I doubt it's expedient for this administration to press the issue.
I haven't seen this; what does it refer to?