The geometry in your picture is too classical. Once you pass the event horizon, it doesn't look like a sphere surrounding you anymore, and you don't see it as a special surface anyway. If you look back along a radial direction, you will see the same horizon point ahead of you (in the past) and behind you (also in the past), at different affine parameter along the horizon (this is clear in a Penrose diagram). But you won't see the horizon as a sphere.
When you approach a Schwarzschild singularity, there is no way to avoid getting compressed to oblivion, because all the volume you carry is compressed to a tiny volume near r=0. The radial area is r, and the area of a sphere is $4\pi r^2$ always, and r is time inside the horizon, and you are necessarily drawn to r=0, which is the singularity. You can't save yourself by conformal mapping, because the actual physical distances are shrunk--- even if you were to conformally get shrunk to zero size, your matter is not conformally invariant, the atoms set a scale.
The dr component of the metric doesn't vanish at the singularity, it's limiting value is ${1\over 2m}$. This means that you are losing a certain unit of r per unit time as you fall in, which means your radial volume is shrinking to zero quadratically with time. The time part of the metric (which is spatial now) goes to ${2m\over r}$, and so you gain a linearly diverging space in exchange, but the quadratic compression doesn't make up in volume for the quadratic sphere shrinking. Further, this is not a conformal transformation in any reasonable sense, it's spaghettification.
The real caveat about black holes is that this whole story assumes the black hole is neutral and nonspinning. For spinning or charged black holes, the interior structure is altered in radical ways, and there is nothing classically wrong with going in and coming out, except for some dubious arguments about what happens when you hit the Cauchy horizon in the interior.
You refer to the "central singularity," but the singularity of a Schwarzschild black hole is not a point at the center of the event horizon. It's a spacelike surface that is in the future of all observers. It's also not a point. See Is a black hole singularity a single point? .
The question you ask doesn't have a meaningful answer. From a point on the horizon, you can draw a null geodesic that intersects the singularity, and its metric length is zero. You can also draw a timelike geodesic, in which case the metric length will be (for +--- signature), a positive real number of order M in geometrized units. You can also draw a spacelike curve whose length in this metric is an imaginary number.
You refer to "proper distance," but that doesn't succeed in resolving this ambiguity. Proper distance is distance defined by a ruler at rest relative to the thing being measured. Inside the horizon, we can't have a ruler at rest. The spacetime inside the horizon is not static.
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The general understanding is that inside a black hole space and and time change their places, see for example explanation here: https://www.einstein-online.info/en/spotlight/changing_places/ . If I understand it right, one should correctly speak about distance and not time for reaching the singularity. Another interesting and unconventional explanation I have heard is that if you crossed (without noticing) by car the event horizon on your Monday, then no matter where you go, how you drive, cross and across, back and forth, it comes Friday and you cease to exist in the singularity. It came me in mind that it is the best description of live, too. You are born without noticing, do a lot of things, driving, too, and on some day you cease to exist. Nice, isn't it?
A more enlightening explanation I have heard from Gerard t'Hooft is:
... . "An exact solution helps: consider a black hole formed by matter that goes in by the speed of light. Doesn't change the physics very much but makes it easy to understand. If all particles (basically without rest mass) would enter in a spherically symmetric mode then you can write the solution exactly. One finds that the horizon already opens up at a space-time point at the center (but no singularity there or anywhere else). As soon as the matter passed the horizon the outside world is in the Schwarzschild metric. Now you have to understand that in the inside region, surrounded by the horizon, space and time interchange roles. What you thought is space (such as the r coordinate) is actually time and what you thought to be time (the t coordinate) is actually space. The singularity is at r=0 but that is actually in the future. Not only that, it is, in a sense, the infinite future because outside observers will never see anything that has passed the horizon. For the outside observer, that never happens. For a black hole formed by matter, there is no past singularity. For quantum mechanics however, everything has to be reformulated. Singularities disappear or become physically immaterial. There are many more such things that people fail to understand, while it isn't difficult.
G. 't Hooft"
I hope, I could help.