Throwing a micro black hole into the sun: does it collapse into a black hole or does it result in a supernova?
The micro black hole would be unable to accrete very quickly at all due to intense radiation pressure.
The intense Hawking radiation would have an luminosity of $3.6 \times 10^{14}$ W, and a roughly isotropic flux at the event horizon of $\sim 10^{48}$ W m$^{-2}$.
The Eddington limit for such an object is only $6 \times 10^{9}$ W. In other words, at this luminosity (or above), the accretion stalls as matter is driven away by radiation pressure. There is no way that any matter from the Sun would get anywhere near the event horizon. If the black hole was rotating close to the maximum possible then the Hawking radiation would be suppressed and accretion at the Eddington rate would be allowed. But this would then drop the black hole below its maximum spin rate, leading to swiftly increasing Hawking radiation again.
As the black hole evaporates, the luminosity increases, so the accretion problem could only become more severe. The black hole will entirely evaporate in about 2000 years. Its final seconds would minutely increase the amount of power generated inside the Sun, but assuming that the ultra-high energy gamma rays thermalised, this would be undetectable.
EDIT: The Eddington limit may not be the appropriate number to consider, since we might think that the external pressure of gas inside the Sun might be capable of squeezing material into the black hole. The usual Eddington limit is calculated assuming that the gas pressure is small compared with the radiation pressure. And indeed that is probably the case here. The gas pressure inside the Sun is $2.6 \times 10^{16}$ Pa. The outward radiation pressure near the event horizon would be $\sim 10^{40}$ Pa. The problem is that the length scales are so small here that it is unclear to me that these classical arguments will work at all. However, even if we were to go for a more macroscopic 1 micron from the black hole, the radiation pressure still significantly exceeds the external gas pressure.
Short answer: we wouldn't even notice - nothing would happen.
Bonus Question: The answer to this is it doesn't have a bearing on the supernova rate, because the mechanism wouldn't cause supernovae. Even if the black hole were more massive and could grow, the growth rate would be slow and no explosive nucleosynthesis would occur because the gas would not be dense enough to be degenerate.
Things change in a degenerate white dwarf, where the enhanced temperatures around an accreting mini-black hole could set off runaway thermonuclear fusion of carbon, since the pressure in a degenerate gas is largely independent of temperature. This possibility has been explored by Graham et al (2015) (thanks Timmy), who indeed conclude that type Ia supernova rates could constrain the density of micro black holes in the range $10^{16}$ to $10^{21}$ kg.
The intense flux of Hawking radiation of about $10^{13}$ Watt will prevent any solar matter from coming close to the event horizon. So, the Hawking radiation creates a small bubble preventing it from growing by accretion.
This might help: http://xaonon.dyndns.org/hawking/
10^9 KG gives it:
a temperature of 1.227203e+14 Kelvin
and a luminosity of 3.563442e+14 watts
and a size about 500 times smaller than a proton by radius - that would make an absorption rate equivalent to its Hawking radiation pretty difficult because it's over five orders of magnitude hotter than the inside of the sun and at the same time, much smaller than an atom.
At that mass, a black hole wouldn't even create a good pocket of very dense material gravitationally pulled material around it. At just the distance of one atomic radius, even in the densely packed center of the sun, its gravity would drop off well over a million fold.
At that size, it's hard to imagine that it would have even significant tidal effects either. If such a black hole existed and you were able to approach it (ignoring the Hawking radiation it shoots out), you'd have to get about 3 inches from it to even feel a 1 G force from it - which would feel strange because the tidal forces would drop off the gravitation rapidly, but as long as you kept a reasonable distance, it wouldn't feel dangerous - perhaps like what it feels like holding a magnet, but you're the magnet.
Now if it was to pass through you it would likely leave a bullet sized hole - so that wouldn't be fun - and its radiation would also be lethal, but if you keep your distance, it would seem gravitationally pretty wimpy until you were very close.
So, if you want a black hole that would eat the sun, I think you have to go bigger - as a ballpark guess, maybe 10^13 or 10^14 kg - give or take and even then, I expect it would take a long time to eat the sun.
Now as to eating the core leading to collapse, a black hole that small wouldn't have a noticeable effect, but as it gets bigger, two things would happen.
It could create a small area of higher pressure, essentially an accretion disk inside the sun and, the formation of the accretion disk would create additional heat as well as those lovely jets that shoot out the poles. The extra heat would likely push matter away from the center of the sun faster than the pocket of high gravity would drag things towards it. The net effect would be complicated because in the localized area you'd have more energy, but that more energy would heat up the sun, causing the sun to expand. It would also have a stirring effect of sorts from the jets of energy. The total effect is, for me, very hard to say.
Now, as the micro black hole gets bigger, the sun would eventually look less and less like a sun and more and more like an accretion disk with two jets shooting out. The intermediate stages are complicated, but the beginning (not much difference) and end (black hole accretion disk) aren't hard to predict.
Now, on going supernova, that, I don't think so because black holes, while eating, shoot out too much heat in the process. A star goes nova because the core cools and in cooling it collapses and in collapsing - well, you know the rest. A black hole would provide steady and consistent heat while it eats, so I see no mechanism for a nova moment - and that's basically how a nova works - it happens kind of all at once. A nova is like a perfect storm, where, everything falls in very fast and then all that matter bounces off of itself and explodes outwards. A core collapse is a very different event than a black hole with an accretion disk.
Maybe I missed something, but that's my take on this rather improbable scenario, and for the record, I don't believe micro black holes exist.