Are any C++ operator overloads provided automatically based on others?

In the core language the various operators are independent. Some are defined in terms of others, but if overload resolution for an operator invocation fails then there is no attempt to express that invocation in terms of other operators. When that's desired it can easily be expressed by the programmer (the opposite, turning off such machinery, would probably be more difficult).

There is a set of relational operator overloads in std::rel_ops that client code can use, defined in terms of < and ==.

You can easily write a mixin-class that provides relational operators in terms of < and ==, or in terms of a tri-valued compare function. That was the original motivation for the Curiously Recurring Template Pattern, called the Barton-Nackman trick.

No.

C++ has no inference rules in the core language, so even defining say + it doesn't assume anything about +=... they're just (as far as the language goes) totally unrelated.

Consider that the << (left bit-shift operator) in the standard library has been overloaded to mean "output to stream"... just because of the look and of a sensible priority and associativity.