Are there any announcements that a captain of an aircraft must make that is reinforced by law or a rule? (This is concerning all countries)
The US does have specific regulations that require the pilot to ensure that passengers are briefed on certain safety issues, although there's no requirement that the pilot must do it personally.
Here are two regulations from the part 91 (all flights) and part 121 (scheduled airline flights) regulations. There may be others.
No pilot may cause to be moved on the surface, take off, or land a U.S.-registered civil aircraft [...] unless the pilot in command of that aircraft ensures that each person on board has been notified to fasten his or her safety belt and, if installed, his or her shoulder harness.
Each certificate holder operating a passenger-carrying airplane shall insure that all passengers are orally briefed by the appropriate crewmember as follows [...]
I guess that other countries have similar regulations, but it would be difficult to list them all, so hopefully this example is helpful for you.
ICAO Annex 6 covers the necessary content on briefings for international flights, and most countries have regulations that mirror those rules for flights within that country. The pilot-in-command (what airlines call the captain) is responsible for briefing all passengers on where to find and how to use:
- the emergency exits,
- the oxygen equipment and life jackets, if they're required for the flight (which they are on airline flights),
- the safety belts or harnesses, and
- passenger briefing cards, if carried;
and for ensuring that all passengers are familiar with the location and use of emergency equipment carried for collective use, for example life rafts or escape slides.
That's all that's required worldwide. Although it's the PIC's responsibility, there's nothing wrong with them delegating to other crew, and this is common practice on airline flights.
The airline will also have a standard operating procedure (SOP), and this may specify additional things the PIC has to say to passengers. Although the SOP isn't a law or regulation, if there's an accident in the flight, and it is found to have been caused by the PIC not following the SOP, the PIC will likely be blamed and this will probably cause them legal problems afterwards.
Only if there is an emergency condition and the captain must tell the flight attendants and passengers to “Brace!” Of course, a pilot may choose not to tell the passengers.
All other announcements are up to the captain. Airlines request that welcome announcements be made, but it remains up to the captain. (source: USA - Today - Travel)