The primary stuff you need to form stars is hydrogen and helium which is absolutely everywhere in the universe. In order for stars to form, however, you need a lot of it. This usually happens if you have a strong gravitational force, which pulls in more and more gas (very very slowly). If the gas cools sufficiently, at some point the gravitational force will overcome the thermal pressure of the gas, causing it to collapse and form stars.

This is the basic mechanism of star formation everywhere and it probably isn't much different in this particular system. The strongest gravitational forces are caused by dark matter haloes which are usually larger than the galaxy itself and much heavier. On smaller scales, the main gravitational force is caused by the collapsing gas itself. Both probably still exist in this system and provide the required gravity for stars to form.

A note: While dust has an impact on star formation (it might help the gas to cool, for example) it is not a required ingredient for stars to form. In fact, the first stars formed at a time when there was no dust in the universe.

I don't know what the black hole is moving through. That is, I don't know how much "metallic" stuff there is in the cloud. The article says the stars are blue, so presumably large and hot. The article also says the triggering event probably happened about 50 million years ago - so after the death of the dinosaurs on Earth.

The article doesn't say how far away this thing is. If it's very distant, and the Universe we're looking at is young, presumably the gas is mainly hydrogen and helium. Maybe, given another billion years or so, the newly-formed stars will coalesce into a new galaxy?

Discarded material from the accretion disc, or (well, more likely) material that was more loosely gravitationally bound.

A galactic core has a lot of mass that could be stolen.

I had the same question! From what I can gather, the black hole is plowing through gas, when the gas gets too close to the black hole, it gets sucked in and forms a disk around the black hole. As the gas disk spirals in towards the black hole, it gets heated up and shocked, causing it to fragment and form clumps. These clumps eventually collapse under their own gravity, forming stars. How cool is that!