Let's say you created a cannon that can shoot small black holes and you shoot it at some star.
Would the star just turn into a black hole silently? Or rather first destabilize and produce a last dying blast like supernovas do?
For high enough speed, the black hole would just pop through the star and fly away. How fast would it need to go for that?
(Asking for a friend, I don't actually have a black hole cannon.)
How much matter can a small black hole absorb? This determines the answer to your questions.
A simple approximation is that moving black holes absorb everything they pass through out to a radius $kR_S$ where $k>1$ determines how widely they can grab passing matter. For a pretty fast hole this is likely less than 2. $R_S$ is the Schwarschild radius of the hole. So if a hole shoots through matter of density $\rho$ for a distance $d$, it will absorb about $\Delta M = \pi \rho d k^2 R_S^2$ kilograms. If a black hole starts with velocity $v$ it will have momentum $v_1M$ at the start, and at the end the momentum $v_2 (M+\Delta M)$ must be the same, leading to an exit velocity $v_2 = v_1 (M/(M+\Delta M))$. For that exit velocity to be above solar escape velocity $v_{esc}=\sqrt{2GM_\odot/R_\odot}$ (about 600 km/s) we must have $v_1/v_{esc}>1+\frac{\Delta M}{M}$, or $$v_1/v_{esc} > 1+ 8 \pi G^2 \rho R_\odot k^2 M/c^4.$$ This tells us a few things. Just dropping a black hole into the sun will make $v_1/v_{esc}=1$, and it will stay. $8\pi G^2 \rho R_\odot k^2/c^4\approx 5.4\cdot 10^{-41}$: the amount of matter a black hole will accumulate when passing through is really small.
What about a black hole just sitting still inside the sun? It will attract nearby matter, but as it gets compressed it will also heat up and the accretion will tend to push away the matter. The limit when the infalling matter is in equilibrium with the pressure is roughly $(1/12)\dot{M}c^2 = 3.4\cdot 10^4 (M/M_\odot)L_\odot$, or $\dot{M}=1.5\cdot10^{32}(M/M_\odot)$ kg/s (assuming thin disk, pure ionized hydrogen and a lot of other stuff which can be debated, especially for big holes). The solar mass is $M_\odot=2\cdot 10^{30}$.
So what this tells us is that if the black hole has a small mass, up to an Earth mass, the amount of absorbed matter is minuscule: there is a hot pinpoint of accretion inside the sun's core, but nothing much changes. Bigger holes will noticeably heat up the core and no doubt have effects on the total brightness. To actually eat the star quickly it needs to have a mass that gets within a percent of the entire star. (Except at that point the above formula doesn't work that well: the star literally cannot free-fall that fast into the black hole.) At this end the final implosion will be pretty bright since the infalling gas get super-compressed and heated, making it produce a bright blast.
Generally, black hole cannons are surprisingly useless.
@AndersSandberg's answer surprised me so I wanted to poke around some more, to see what could a determined attacker achieve. So here's what would happen if we put a BH in orbit inside a star so that it can continuously feed.
Let's use our Sun as an example, with densities (raw data):
For a given $r$, our orbital speed is $v_o\approx \sqrt{\frac{GM_{below}(r)}{r}}$, and mass eaten per $s$ per $m^2$ is $v_o\cdot density$, which gives:
BH will eat the fastest when orbiting at $9\%R_{Sun}$. Below is a simulation of that. I assumed $k=2$ and a BH mass of $13M_{Jupiter}$ (that's the mass of the heaviest known planet). (BH not to scale.)
We can also try just dropping the BH, so that it passes through the dense core. After trial and error, dropping from $\approx 20\%R_{Sun}$ seems best, but not much better than the orbit:
For trajectories in between these two, eating speeds are similar. But if we don't manage to put the BH on such a narrow trajectory, the eating speed is much worse:
So for our highest eating speed of $6\cdot 10^{14} kg/s$ it will take about 1 milion years to eat 1% of the Sun's mass. And we used a BH with mass $13M_{Jupiter}$. Eating speed is proportional to $R_{S}^2$ which itself is proportional to mass. So for 10x smaller BH, it feeds 100x slower, which gets hopelessly slow. That square relation also means that shooting many smaller BHs is not worth it - better to use one but big.
But you may have noticed that our $13M_{Jupiter}$ BH is already above 1% of Sun's mass (which Anders proposed as the point where we're getting close to explosion). So even for this already explosive BH that doesn't need to grow, if it was growing through movement, that would still be extremely slow. If we instead used a Jupyter mass BH, it would need to feed for about 150 mln years before it gets explosive.
Some things that may boost the eating rate:
Some other things I ignored:
To kill a star, you'd need to throw in a BH that's already huge (which is possible but really expensive). Smaller BHs will just result in star poisoning, shortening its lifetime, although it's not clear by how much. And the hosts may try to scoop that BH out (which would be the ultimate game of curling).
If the BH heats up the star a lot without destabilizing it, there could be cases where you want to do it intentionally to your own star, to "overclock" it. Later you can also scoop out that BH to go back to normal.
The existing answers haven't really addressed your point 2, so I'll cover that here.
The possibly-surprising thing is that it's basically impossible to shoot a black hole at a star such that it won't go straight through. The reason is that if you're shooting it from outside the star system then it's on a hyperbolic orbit, or at best a parabolic one. In other words, even at the slowest possible speed it's falling into the system from very far away, getting faster and faster as it reaches the star, so that when it gets to the star it's picked up exactly enough speed to escape again. The only way to avoid that would be through gravitational slingshot type interactions with planets, but it will be very difficult to set those up so that the black hole gets captured. (Or by friction as it passes through the star. But as Filip Sondej mentions that's very small for a black hole of with enough mass to actually eat the star - it will barely even slow down.)
Even if your black hole gun is located inside the system you still have a similar problem. You can fire the black hole onto an elliptical orbit that passes through the star, but it's still going to pass through it and continue on its orbit, spending only a small fraction of each orbit inside the star. The orbit will slowly decay until it's entirely inside the star, but that will take a very long time.
All of the answers here deal with the black hole eating matter as if nothing else happens in the process.
This is plain wrong.
Black holes (well, it is the space neighboring them) are the brightest objects in the astronomy.
Black holes convert between 5% and 40% of the infalling mass into light.
This is how a stellar black hole can outshine an orbiting star, capturing just a small part of its stellar wind. This is also how a theoreticized quazi-stars could exist.
In short, a black hole eating thru a star would create an immense explosion, comparable to supernovas (at least some of them are powered by similar mechanism), possibly completely destroying the star rather quickly.
In the process, only a small part of the stellar mass will sink into the black hole. Most of it will be blown away.
