I recall an anecdote about a mathematician being asked to clarify precisely what he meant by “a close approximation to three”. After thinking for a moment, he replied “any real number other than three”.

“Approximately equal” is just a superset of “equal” that also includes values “acceptably close” (using whatever definition you set for acceptable).

Unless you say something like:

a ≈ b ∧ a ≠ b

which implies a is close to b but not exactly equal to b, it’s safe to presume that a ≈ b includes the possibility that a = b.

It depends on the convention that you use, but in my experience yes; for any equivalence relation, and any metric of “approximate” within the context of that relation, A=B implies A≈B.

Yes, informally in the sense that the error between the two numbers is “arbitrarily small”. Sometimes in introductory real analysis courses you see an exercise like: “prove if x, y are real numbers such that x=y, then for any real epsilon > 0 we have |x - y| < epsilon.” Which is a more rigorous way to say roughly the same thing. Going back to informality, if you give any required degree of accuracy (epsilon), then the error between x and y (which are the same number), is less than your required degree of accuracy