Yes, this is common practice in some countries. Many people 'oil' or 'grease' the palm of those who have a position of authority in the country (= they bribe them) in order to achieve their goals. Therefore, both sides behave in an illegal or dishonest way...

It means "to bribe" - but there's little explanation on the analogy, so here's my thoughts: The palm (of the hand) is where you receive money, of course. You grease (or oil) something to make it work smoother. So you grease someone's palm so they can work better for you, to your advantage. And that "grease" is money.

it means to give money to someone in authority in order to persuade them to do something for you, especially something wrong. cowboy...

Yes it's a bribe either in the shape of a gift or money.

It's to pay a bribe in the hope of a favourable outcome