What can the US House do without approval/involvement of the Senate?

Yup, this.

Repealing the ACA, repealing pre-existing condition coverage, tax cuts for the rich, etc. will not happen under a democratic house (at least I hope not).

Also they can hold investigations.

I believe that is it.

The senate on the other hand, can do 3 things while the house can only do 2 (assuming the dems held the senate but GOP held the house). In that situation the dems could.

[ul]
[li]Block legislation[/li][li]Hold investigation[/li][li]block judicial nominations[/li][/ul]

Blocking judicial nominations would require a majority in the senate, but the house has nothing to do with that.

I don’t know if there is anything else the house does itself, that they don’t need the senate for.

Couldn’t they investigate the Kavanaugh claims in private, and then either release the findings or turn them over to the justice department?

If they investigate in private, but then release the findings I think the public would respond better to that.

It’s mostly meaningless. Just because budget bills have to originate in the House doesn’t mean much in the end.

Same thing as always. Impeach civil (and judicial) officers, (theoretically) control the initiation of revenue bills and elect our next President if the EC can’t.

a member of the house is an expert on impeachment since he was impeached and removed as a federal judge. He’s been in the house since 1993
Alcee Hastings - Wikipedia

They also don’t have close to a veto proof majority so Trump can just veto any bill he doesn’t like.

Besides which, any member of Congress can introduce a bill. Even the Republicans aren’t so incompetent that they couldn’t find one single Republican Representative to introduce whatever spending bill they wanted.

It would stay “private” for 5 seconds, maybe less.

Unless Mueller were doing it.

Weten’t you the one asking what the House can do? Is Mueller part of the House? Was the post I responded to about Mueller or the House?

I guess I should have made my point more clearly and exposed all the links in my chain of thought. You’re right. It would not stay private in the House. In order for it to stay private, such an investigation would need to be handled by an independent investigator who has proven that he can control leaks. Someone like Mueller. Mueller is not a member of the House of Representatives. Quod erat demonstrandum. Such an investigation If carried out by the House would not remain private.

I explain the House’s power to those around here as สิทธิบังคับพยาน which might translate as the Right to Compel Testimony.

But that leads to the question: How will they compel testimony? I fully expect Trump and his fellow criminals to smirk and say No. Will the United States Marshall Service help out? They report to the criminal Matt Whittaker. In January the House will appoint a Sergeant at Arms of the United States House of Representatives; perhaps they should appoint several hundred of them and equip them with assault rifles. The Metropolitan Police Department of D.C. reports to the Mayor of D.C. (unless POTUS declares an emergency) — will they ally with the House and help arrest subpoenaed witnesses in contempt?

Strongmen do not give up power easily. Many Republicans have lost all respect for the Constitution. Treacherous times may lie ahead.

That doesn’t mean it ever comes to a floor vote. The Speaker and the committee heads decide what bills move forward.