Are there any mandatory announcements that a captain has to make?
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.

14 CFR 91.107(a)(2):


  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.


14 CFR 121.571:


  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.