Is 0.0.1 valid semver?

There are no rules here and conventions are loose.

When the initial major number is zero, all that is guaranteed is that higher numbers come after lower ones. All the semantic versioning guarantees about "this is a bug fix" or "old functionality still works" only apply to version one or later.

While there is a loose convention of 0.1.0 being the lowest version number, there are those that would argue it the first version one shares with others. The specification of semantic versioning simply sees the leading zero and washes its hands.

The semver 2.0.0 specification doesn't preclude it. The FAQ does recommend starting at 0.1.0 though.

How should I deal with revisions in the 0.y.z initial development phase?

The simplest thing to do is start your initial development release at 0.1.0 and then increment the minor version for each subsequent release.

As per the semver calculator (https://semver.npmjs.com/), it is not recognizing the versions which start with 0.0.x.Semver Calculator Screenshot