Reversible processes are important because they are related to the efficiency of a process. Take for examples a pair hotplates, one at 100C and one at 0C. In a theoretically ideal setting you could extract some work $W_0$ from this system until the two hotplates reached equilibrium. Then we would say the process is reversible, because in the same ideal world you could input $W_0$ into the system and once again create the same original temperature gradient in the hotplates.
In the real world, because of friction, poor design, and other factors, you might only extract $.7W_0$. We would say that some work in the system is lost to irreversible entropy. This is entropy which can't be reversed unless you add in extra work from the outside. You can partially reverse this system, but because you weren't able to extract all the possible work, you won't be able to reach the original temperature gradient. You might only get one hotplate to 80C and the other to 20C.
In the worst case, you simply touch the two hotplates together. In this case you can't extract any work from the system, and all potential work is lost to irreversible entropy. There is nothing you can do to restore the system to it's original state without doing extra work on the system added from the outside. So to more specifically answer your questions:
2) Reversible work and heat flow are just the parts of a system that make sense when you run it in reverse. So for example, compressing a gas with a piston make sense in reverse. But touching two hotplates together does not make sense in reverse.
1) The significance of reversibility versus irreversibility is related to efficiency. We can't extract work from an irreversible process. So systems like engines are designed in such a way to minimize the irreversible entropy and heat flow.