Do children processes inherit ionice priorities from their parents? How do you check the IO priority of a running process?

Yes. I tested it. IO priority is inherited just like CPU niceness. This is probably what you want. If it isn't, you can explicitly specify the IO priority of child processes with the ionice command.

Any processes that are forked inherit the io class and priority of their parent, however an important fact to keep in mind is that if you alter the io class and/or priority of a running process it doesn't change the io class/priority of any current child processes of that process. That said, once you have adjusted the class and/or priority of a running process, any child processes that are forked going forward will inherit the new io class and priority.