To make an angry protest against a relatively minor problem, in the process embarrassing the protester. The analogy is with a baby who throws toys out of the pram in order to get their parent to pay attention to them. The implication in the idiom is that the protester is acting like a baby.
If something is to little avail, it means that, despite great efforts, something ended in failure, but taking comfort from the knowledge that nothing else could have been done to avert or avoid the result.
If a ship sails under a flag of convenience, it is registered in a country where taxes, etc, are lower than in the country it comes from, so if someone does something under a flag of convenience, they attempt to avoid regulations and taxes by a similar means.