Is what the questioner ate of the oil that his family received as a gift considered a bribe, and is he blameworthy for eating it despite his attempt to vomit? And is his drinking of the free beverage in the internet cafe, which is frequented by patrons of forbidden websites, considered consuming something unlawful?

If the gift is in exchange for intercession or is a bribe, then it is illicit and forbidden. The Prophet, peace and blessings be upon him, cursed the briber, the bribed, and the intermediary. As long as you consider what reached you a bribe, then trying to regurgitate it is the correct action. As for drinking tea in a cafe, if it is not known that its price is from ill-gotten gains, then there is no sin in it. However, if it is known to be from ill-gotten gains, then it is forbidden.