Do black holes gain mass when matter falls into them?

Yes. (And the radius of the event horizon grows proportionally.)

black holes also radiat mass, though they normally do it at a much smaller rate than they gain mass.

Most of the matter falling into a black hole never enters the event horizon, but rather gets blasted away in polar jets. Black holes are very messy eaters.

Following the laws of physics 'I'm assuming', yes, they do. Matter is conserved, so if it is 'sucked in' it would stay in there meaning the black hole has more mass. This gets complicated very quickly, but surely the answer is yes?

Most of the matter falling into a black hole never enters the event horizon, but rather gets blasted away in polar jets. Black holes are very messy eaters.

How does this matter escape if nothing can escape a black hole?

As long as the matter is outside the event horizon it is still free to go on its merry way.

So the jets that they are talking about are outside of the event horizon, correct.

Correct