could a spacecraft slingshot around a black hole without being pulled into it?

Yes, but you need to use General Relativity, not Newtonian physics, to do the calculations.

It turns out that the ship can not get closer than 3 times the radius of the event horizon and escape if it is on an unpowered orbit. If it was closer than 3 radii, it would need to turn on its engines in order to escape. Closer than 1 event horizon radius and it will be impossible to escape.

It depends on the height of the slingshot. In principle a Black Hole can be treated as a large mass in a small volume - say three Solar Masses - easy to go around.

If mass is actively falling in to the event horizon then the radiation levels are extreme. If there is no mass falling in the radiation level is nill.

There IS an orbital height below which the orbit cannot be predicted and tidal disruption is possible - but in general it is just another mass in space.

Those are good answers, except for super massive black holes. A sling shot maneuver as close as a mile from the mass center of a two solar mass black hole, may be useful, except colliding with very fast subatomic particles is bad and the craft needs to be sturdy to avoid tide stress damage. Sling shot maneuvers can slow or speed up a craft and change the direction of travel. Likely some black holes have negligible accretion disks and negligible polar jets.

Possibly...

But its electronics might be fried by radiation.

And it might be pulled apart by differential gravity. -- the side of the spacecraft nearer to the black hole feels more gravity than the far side. The difference in gravity will pull it to pieces if it approaches to close.

If the distance was such that the spacecraft survived, it would receive an enormous boost of speed. Like what happens when Voyager 1 and New Horizons got a boost of speed when they passed near to the planet Jupiter.

Yes, as long as it didn't go within a certain distance of the event horizon. Black holes have 'jets' of materials that seem to be 'ejected' from the area, caused by gravitation interaction of material that did not cross that point of no return.

Once breached, light can't even escape.

Yes, a black hole is no different than any other gravitational source of the same mass. The only difference is that you can get much closer to one. But just so long as you're outside of its event horizon, the possibility of escaping it is more than zero.

Yes. It would just have to be very far away haha But you can get in orbit or slingshot around a blackhole, yes.

If it maintains a safe distance from the event horizon & have enough velocity.

Yes of course it could,

Unless that spacecraft moves faster than the speed of light