Q1: For example, do banks in Ecuador and Panamá have accounts/reserves in the US Federal Reserve?
No they don't only depository institutions that fall under Federal Reserve Act [FRA] are eligible, and FRA only applies to US banks (Federal Reserve of San Francisco).
Q2: Is a USD bank deposit in Ecuador equivalent to an USD deposit in the US? Is it possible to transfer USD of a bank in Ecuador to a bank in the US at face value?
Yes and no. Nominally yes it is completely equivalent, \$1 bill is \$1 bill everywhere in the world.
But the real purchasing money differs. So in terms of purchasing power they are not equivalent. One dollar does not have the same purchasing power in New York that it has in Kentucky, and one dollar does not have the same value in Panama that it has in US.
Also if your account is not subject to some restrictions, for example your money is not frozen by government because you are suspect in some illegal activity, you can transfer money from Panama to US. You will have to pay bank fees but there is no exchange rate between Panama US dollar and US US dollar.