Am I allowed to videotape (dashcam) when entering the United States by car?

Probably, but you may have to be prepared to litigate. Other answers here which purport to give a definite yes or a definite no are wrong.

The excellent Papers, Please! blog has a detailed writeup of a 2020 settlement between the ACLU and the DHS, in which it was agreed that

[t]he Department of Homeland Security and its components, including but not limited to U.S. Customs and Border Protection and Federal Protective Service, shall not prevent, impede, or otherwise interfere with the First Amendment rights of members of the public to make and retain photographs, video recordings, or other recordings of matters or events from a Publicly Accessible Area at any land port of entry in the United States. A Publicly Accessible Area is an outdoor area at a land port of entry where a member of the public is permitted to be regardless of whether they intend to cross the United States border.

Papers, Please! adds that

We don’t mean to suggest that you don’t also have the right to record or livestream what happens to you as you cross the border. We think you do, but that hasn’t yet been litigated as extensively.

The ACLU has a form to fill out if you believe your rights under this settlement have been violated.

Video recording outward from your dashboard or windshield, or rearward from your back window

No problem at all in the US. You are recording scenes in which noone has any expectation of privacy, specifically a public street or highway.

Audio recording the conversation that takes place inside your car?

This gets iffy very quickly. A significant portion of US states make this illegal. Check the laws in each state you plan to travel in. Helpful search terms might be "one party consent" and "two party consent" to recording a conversation.

For this reason many dash cam makers that market their products in the US make it very quick and easy to start/stop the audio recording function. I myself own Garmin front and rear cams, and I can start or stop audio recording on both cams instantly with a single button click. I do audio when I’m alone in the car, audio-off when I have passengers.

Rear facing camera mounted in the front?

This situation I really don’t know about. It’s extremely widespread amongst taxi drivers, fleet drivers, and ride-sharing drivers, so I assume it’s legal in at least some cases. But there might be some requirement to inform passengers with a sign or something.

While crossing the border

I think it’s highly unlikely that any border officer in the US would take any issue with camera(s). If they do, then of course follow their commands at the time, whether it’s to unplug the cam or completely pull it off your windshield. Then you can restart the camera(s) immediately after leaving their presence. Their professional opinion of the law is frequently incorrect and won’t hold water in court. But it’s better to defuse the situation long before a court appearance is necessary. If they can make a case that the camera(s) is obstructing the driver’s view you will likely lose that case.

Recording in restricted areas

There are some places where video recording is prohibited. Military bases. Commercial ports. Maybe also international border crossings?

I’d like to think the border crossing officers are enlightened enough to realize that any car with self driving technology is probably recording extensively and that recording cannot be stopped. Teslas do for sure, and probably also many other high end high tech cars.

Bottom line

What I would do is pop that camera off the window and throw it in the cupholder just before approaching the border. Immediately after crossing, you can pull over and put the camera back up.

There are some states that have laws against recording people, and these laws can depend on how close you are to the person being recorded, whether you have their permission, and whether audio is being recorded. Border agents are federal employees, so they probably won’t be too concerned by state laws, but they theoretically could enforce them.

I don’t think there’s any federal law against recording. Even if there are, there is a strong constitutional case against such laws, as well as the state laws. But from a practical perspective, even if there is no statutory/constitutional basis for the prohibition, there’s a lot that CBP can do to make your life more difficult. And while the constitution does apply to non-citizens, it’s more difficult for them to assert it.

A quick google search yields this: https://clearit.ca/canadian-customs-broker-blog/dashcam-recording-prohibited-u-s-ports-entry/ and this https://www.immigroup.com/topics/it-legal-use-dash-cam-border/ etc.

Summary: As is typical with the USBC, it’s "we do whatever we want whenever we feel like it" or more politically correct: "the officer has the discretion to allow or disallow it". The legal underpinning to disallow it is this: https://www.law.cornell.edu/cfr/text/41/102-74.420 and technically you would need permission or even written permission.

Enforcement seems to be all over the place, some people got significant amount of harassment for having a dash cam, but many other people report no problem whatsoever.

It’s legal and allowed on the Canadian side.