>If the speed of light of this universe was raised significantly, is there some point where black holes would be impossible to form?
If the speed of light was infinite, yes, but a lot of physics relies on the speed of light being finite. Pretty much everything breaks (e.g. mass) if you have an infinite speed of light.
>I assume if the speed of light were lowered black holes would become far more common?
Yep. It might clarify things a bit to say that the "speed of light" is a misleading name - it's really more about being the speed of causality, the speed at which things happen.
In the Death's End by Cixin Liu, there is a speed of light weapon that lowers the local speed of light to almost nothing, effectively taking an enemy out of the game until the end of the universe.
In Redshift Rendesvous (don't recall the author), there was an environment where the speed of light was just above running speed - maybe 15 m/s. You observed relativistic effects just walking around.
Hard to imagine how different the resulting universe might be, what if the speed of light and causality were say 10x higher than what it is now? Or what if the speed gradually increased over time? Would we just evaporate?
I would imagine that a universe that doesn't obey basic causality will not exist for long, even if it comes into existence, nor will it support complex structures that can observe and reflect (like life). Other universes' physics and evolution might be very different, but whatever its laws, on a macro scale I'd imagine they follow internal consistency. So yes, we might evaporate, but it would be well predicted by the science of that universe. It won't happen inexplicably. :-)
I would have thought in such a strange universe there could at least exist some structures capable of knowing joy, but that thought has been thoroughly eradicated.
I assume if the speed of light were lowered black holes would become far more common?