At this point it doesn't even really matter, it's just a question of whether anyone would step up to stop them. "The Rules" in general have already been well and truly trod upon and I'm not sure we make it out the other side of this without some restructuring of them.

Obviously with the Senate they can cut any budget/pass any law.

On their own the house can cut funding eventually. Push comes to shove, the house needs to vote for spending. They either pass a budget with the cuts, or the government shuts down.

There really isn't a mechanism for the house that nationalize anything without the support of the Senate that I'm aware of. The house can call the vote day one to nationalize, but it's symbolic without a legit path on the Senate

Iirc any resolution to that extent would likely have to pass through both levels of Congress. But, if enough seats flip in both chambers next year, it could certainly happen. Checks and balances and whatnot.