A Four-Fold Accumulator is one bet made up of four selections in different events. If they all win, so do you. But if any of the selections lose, your bet does too.
If a bet is on the winner of a game, that is called a moneyline bet. If you're betting that a team will win or lose by a certain amount of points, that is called a spread bet. If you combine multiple outcomes into one bet, that is called a parlay.
No, but there's no reason they would ever kick anyone out for that. Far more likely, the player would get a comp of some kind to encourage them to play longer.
The Martingale system is perfectly legal. Many online casino players use it to determine how much to stake on each hand or spin of the wheel. You can also use the Martingale system in sports betting., but a horse racing Martingale system would be difficult.