The priorities of && || vs. == etc. came about in the following way.
Early C had no separate operators for & and && or | and ||.  
(Got that?)  Instead it used the notion (inherited from
B
and
BCPL)
of "truth-value context": where a Boolean value was expected,
after "if" and "while" and so forth, the & and | operators were interpreted
as && and || are now; in ordinary expressions, the bitwise interpretations
were used.  It worked out pretty well, but was hard to explain.
(There was the notion of "top-level operators" in a truth-value context.)
The precedence of & and | were as they are now.
Primarily at the urging of Alan Snyder, the && and || operators were
added.  This successfully separated the concepts of bitwise operations and
short-circuit Boolean evaluation.  However, I had cold feet about the
precedence problems.  For example, there were lots of programs with
things like
if (a==b & c==d) ...In retrospect it would have been better to go ahead and change the precedence of
& to higher than ==,  but it seemed safer just to split & and &&
without moving & past an existing operator.  (After all, we had several
hundred kilobytes of source code, and maybe 3 installations....)