Suppose a black hole were to suck up a massive amount of surrounding oxygen and hydrogen. Would this ignite inside the black hole and turn it back into a star?
No. The basic point about black holes is that they are black, i.e. nothing escapes from them. Barring Hawking radiation (a quantum effect) there is no way to destroy a black hole. By throwing things into it, you just make it bigger.
Of course, it's perfectly possible for a star to form inside a black hole (at least in principle). This is because big black holes are pretty nicely behaved and if there would be a proto-star dust sucked into them (meaning they would fell under the horizon), it wouldn't experience anything out of ordinary and could form a star. And perhaps, given enough time, also form a black hole of their own.
In short, life under the horizon of the big black hole is in every way normal except that you can't escape from the inside of the horizon and eventually will hit a singularity and be destroyed (that is, if you won't get destroyed sooner by lots of other effects, like tidal forces).
the answers given by Marek and inflector are perfectly fine. Given our knowledge of general relativity and the spacetimes of black holes, there is no escape, no black hole transformation, not even another supermassive black hole would change the situation.
The reason I am writing is that we should be a little more careful. The point is that the laws of nature we are aware of are reliable on energy scales we can access/measure somehow.
Analogously to your question we could ask if there is an "amount of" energy we would need to transform a black hole somehow. And the answer is: We just don't know, we are not aware of some black hole transformation theory :)
No, it would not. An explosion of any power involves matter which has mass and cannot even reach the speed of light. And even the speed of light is not fast enough to escape a black hole. So the explosion would never escape the black hole if it were to occur.
There is no way to turn a black hole into a star. Adding mass just makes it bigger. A little chemical explosion is very little energy compared with the energy of fusion which powers a star. The power of the explosion doesn't matter. Even a supernova inside the event horizon would not turn it into a star.
It is possible that oxygen and hydrogen would burn beyond the event horizon if they were in close enough proximity and temperature, they might even explode. But the byproducts, even the light from the burning or explosion would not escape the black hole's event horizon.
No, matter that has crossed the event horizon can never send any signals to the outer space anymore. In particular, it cannot become a part of a star that could shine (and be seen ouside the black hole). This would violate the very definition of a black hole.
Of course, some hydrogen gas can get very hot while being swallowed to the black hole, but it will stay inside. Once it's inside, the life expectancy is not too high, anyway. A solar-mass black hole has a radius of about a few kilometers, so it takes a microsecond or so for all the matter to be destroyed at the singularity.
Even the galactic-center black hole whose mass is in millions of solar masses only allows the matter inside to live for an extra second (or a few seconds) or so.
Black holes are the final stage of the evolution of any bound system. They have the maximum entropy among all bound objects with the same mass (and charges). The only way how they can evolve is to evaporate by the Hawking radiation - which increases the entropy further. But it's only possible because the outgoing radiation is not bound and the total entropy may still grow.
However, a back-transformation of a part of a black hole to a star would reduce the entropy - because a star has a vastly lower entropy than the same-mass black hole - which would also contradict the second law of thermodynamics.