This is actually two questions. I'll answer them in order.
The answer to the first is depends strongly on exactly what you mean by a black hole. The most commonly used defining feature of a black hole is the existence of an event horizon. However, the definition of an event horizon is very non-local and in particular requires full knowledge of the future of the spacetime. This leads to some peculiar results when try to define "when" the event horizon formed. For example, in the <PERSON> diagram of a collapsing shell, the event horizon appears in the interior (flat) part of the spacetime "before" the shell crosses its own Schwarzschild radius.
A more local definition of the formation of a black hole is the appearance of an apparent horizon, a boundary on a time slice beyond which all light rays "point inwards". Black hole singularity theorems link the formation of an apparent horizon to the formation of a singularity in the future, while the (weak) cosmic censorship conjecture implies that an apparent horizon must always lie inside an event horizon. These points combined make the appearance of an apparent horizon a good indicator for "when" a black hole is formed. 43
This is also what is used for this purpose in most numerical relativity simulations for detecting the formation of black hole horizons.
However, the notion of apparent horizon depends of the time slicing of the spacetime, making their position not fully invariant. (and therefor their location in the <PERSON> diagram not well defined.) However, for somewhat "sane" slicings of a collapsing shell spacetime, the formation of an apparent horizon coincides with the shell crossing its own Schwarzschild radius.
As for the where in the <PERSON> diagram the first <PERSON> radiation is emitted. This is not well-defined within semi-classical gravity. Hawking radiation is derived through an essentially global argument. I.e. its existence is derived only on future null infinity in the <PERSON> diagram. You can trace back the Hawking modes using geometric optics, in which case you will find that (in an eternal BH spacetime) the modes originated inside the white hole region in the past. In a more physically relevant spacetime describing collapse, they will always stay outside the apparent horizon.
To truly say where/when <PERSON> originates one would need a full quantum gravity description of the process, which we currently don't have. 43