general relativity - Does a black hole have enough time to actually form a singularity?

I am trying to wrap my head around black holes, singularties and hawking radiation. Physics.se contains many intresting questions and answers, but from none I could so far read about the interaction between formation of a singularity and hawking radiation (most seem to talk about the point at which matter plunges into a black hole that already has a singularity).

From what I understand, from the perspective of a star collapsing into a singularity, the perceived time is just like in classical mechanics: the mass is accelerated towards the center of gravity, and when "everything" is there, it is a singularity. Now this takes a finite time, and the matter itself perceives just that. Part of the question is: how long is this time? I am assuming in the magnitude of milliseconds.

Now as I understand hawking radiation, as soon as there is an event horizon, it will start emitting negative mass/energy into the direction of the singularity. Time dilation gets infinitely stronger as we approach the singularity, so I would assume that from the perspective of those negative particles, they would all the time reduce their perceived distance to the particles of positive energy/mass that are already on their way to the singularity.

Since approaching the singularity, time dilation becomes kind of infinite, even the $10^{100}$ years or so it is supposed to take for hawking radiation to evaporate a black hole seem to be enough to send the necessary amount of negative energy/mass particles down to the singularity so that they can "chase" the positive energy/mass paticles very closely.

But will they reach them before a singularity forms, so that they cancel out each other to the point there is not enough gravitational force?

If this is just a matter of calculating "$n$s until matter reaches singularity" and "$m$s until negative mass/energy reaches singularity" and calculation shows that always $n

If this is a qualitative matter, in that some of my ideas are fundamentally wrong, please point those out (again if possible in a way that doesn't require too exotic concepts).