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Can I place a stock limit order to buy above the current price? Can I place a stock limit order to sell below the current price?
You can place the orders like you suggested.  This would be useful in a market that is moving quickly where you want to be reasonably sure of execution but don't want the full exposure of a market order.  This won't jump your spot in the queue though in the sense that you won't get ahead of other orders that are "ready" for execution just because you have crossed the spread aggressively.