True/False: In general, a firm's profit is the same regardless of whether it solves a profit maximization or cost minimization problem.

Answer: True. Cost minimization is the "dual problem" of profit maximization. In other words, the maximum profits are the same, whether you use a cost minimization or profit maximization problem.
A detailed proof is probably beyond the scope of most undergraduate courses, but can be found in Mas-Colell, Whinston and Green.

(The output of a cost minimizing and profit maximizing firm need not be the same; consider the case of constant returns to scale.)

True/False: Consider two different firms with identical production functions (but you do not know what the production functions are). The firms are doing static optimization (as opposed to dynamic optimization, where there are many time periods). Suppose one firm chooses to maximize profits and the other firm chooses to minimize costs, subject to producing the profit-maximizing level. Both firms will produce the same output.

Answer: False. Consider the case where both firms have constant returns to scale (CRS) technology, and the zero profit condition holds. Then any amount of output is profit maximizing. For example, one firm could produce 0 units of output, and the other firm could produce 5 units of output, and both firms would have the same profit. Hence the statement is false.