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.
Yes, the Martingale strategy is legal on betting sites. This strategy, which originated in 18th-century France and is commonly used in games of chance that offer a 50/50 bet, involves doubling your bet after each loss so that the first win would recover all previous losses plus win a profit equal to the original stake.