Is there any situation in which a primordial black hole-sized black hole form AFTER the "Big Bang?" I have seen that primordial black holes may have formed at the start of the universe due to the density of matter. So, is there any way in which our current Universe could spawn a non-stellar mass black hole.
 A: I don't think I can explain all the technical stuff, but first things first.   
Primordial means "at the creation", so if it was created today it wouldn't be primordial.
Now "Primordial size" black hole, is probably what you mean and even that is a bit vague as estimates vary on the possible sizes of Primordial black holes, (and there's some uncertainty as to whether they exist at all) . . . but in the spirit of your question, I think you're asking about smaller black holes than stellar mass black holes. 
The answer to that is almost certainly a hard no.   The Universe can't create smaller than stellar mass black holes because too much energy has to be squeezed into too small a space for a that size black hole to happen.     Is it theoretically possible, if we focus a ton of energy into a small area . . . well, yeah, theoretically, but not practically, and I think it's extremely unlikely the Universe can do it today.
A fraction of a trillionth of a second after the big bang when the universe was crazy-dense, then, primordial black holes may have formed in the very dense and not quite uniform soup of matter but there's still a good deal of uncertainty there, not only on whether it happened but on the mass of them if it happened.
https://en.wikipedia.org/wiki/Black_hole#Primordial_black_holes_in_the_Big_Bang
It's possible that in a high energy particle collision micro or quantum black holes can be created (see link above as well), but that's probably very difficult too.
The easiest way to create a black hole is the old fashioned way.  Get at least 5 solar masses together in one place and wait for the fusion energy to stop.   Any less mass than that and it's very difficult, perhaps impossible.

Could those primordial black holes have survived to today? Estimate
  their size on whatever the upper limit estimations are, AND with the
  lower limit estimations.

I want to say first that I'm a hobbyist.  There's way way smarter people here than me, but I can try to answer this as well as I can.
There's a few points to cover.
1) primordial formation (and that's largely theoretical)
2) lifespan of black holes (Here's a convenient link to check the expected lifespan of black holes based on Hawking Radiation (which might not be accurate either, but I think there's more consensus behind that).  
and 3) Observation and observation is important, because, if we can "see it" or see a footprint signature via the Hubble telescope or something else, then that's darn good evidence and that's what they want.   If they see something that can be explained by primordial black holes - that's good info, and a number of physics discovered were made based on observation including some very big discoveries like dark matter, dark energy, confirmation of black holes, hubble expansion and likely more, so observation is really important.
Lets take point 2 first.  A black hole with a lifespan of 13.8 billion years (based on the calculator above) would have a mass of about 170 million tons and it would be smaller than a proton (not much smaller, but a bit smaller, about 1/2 the size).
So, if primordial black holes are capped at a size of 170 million tons in mass, they might all be gone by now, unless they found a way to feed, but being smaller than a proton, that might be unlikely.
If they exist and were larger than 170 million tones at the time of formation, then there's a good chance they're still around.
Switching to point 3 - observation.   (I looked, but couldn't find a good article), but I remember reading about this.   If they're small enough (but larger than 170 million tons at formation), they would be quite hot and we might be able to observe the hawking radiation signature - if they were close enough.
And if they're much larger than that, the hawking radiation might be too low to observe but the gravitational lensing becomes more observable.  I believe I read a question here that said the observation window is closing and primordial black holes don't appear to exist, cause if they did, we'd have observed either hawking radiation or gravitational lensing, so the observational evidence isn't good (but I'm not an expert on that).
Now "maximum size" - jeepers.  The sky is the limit.   Millions, perhaps billions of solar masses.   It's possible that the massive black holes in the center of galaxies are all primordial - now, I'm not saying that's true, only that it's possible.   In a sense, super-massive black holes are easier to create than small ones cause smaller black holes require much much greater density and can only be created when the universe is incredibly young, like a trillionth of a trillionth of a trillionth of a second.   Large black holes can form a fair bit later, but that still doesn't prove they exist.   I think it remains unsettled.   
My opinion - primordial black holes don't exist, or, if they do, they're the super massive ones in the center of galaxies - but take that with a grain of salt.   I'm not an expert.
A: Actually its perfectly possible within standard GR to do so! The question then becomes at what point does GR break down, and if it formed a mass at that point would it have time to decay via hawking radiation down to a Planck remnant? 
Many types of matter exhibit what is called type II critical collapse where if you carefully tune initial data, you can create an arbitrarily small black hole. In particular, this is true of the radiation fluid which is the limiting case of an ultra relativistic perfect fluid. Of course when you get to very high densities the state equations would change and your self similar collapsing solution would no longer be a solution for the new type of matter. Regardless, it should be possible to choose initial data such that this new phase of matter evolves to its type II critical solution as well. 
A: If people brought all the trash in the galaxy to a single place and dumped it, a black hole could eventually form from the ever-increasing density without ever forming a star. Hypothetically, a bunch of heavier elements (incapable of starting a fusion/fission process to hold the gravitational collapse at bay) could happen to end up in the same region of space and just collapse into a black hole.
I don't think either of those scenarios is very likely though.
