If price is above AVC then not in the short run. By shutting down a firm will incur only fixed costs. If price is above AVC the firm will cover all variable costs and at least some fixed costs. At this price, the firm is earning a normal profit. It assumes technology and labor skill levels are held constant.
Marginal cost is the change in total cost due to a unit change in output.
Fixed costs are independent of output and therefore not part of marginal cost. When it is above the average, the average is rising. Remember the example of the marginal and the average exam grades. In order for the average exam grade to increase, the marginal exam grade must be above the average. The opposite for the average to fall.