Perhaps the difference lies in the distinct meaning or the words in question. For the problem statement, state the problem. For the mission statement, state your mission, or how you will address the problem.
The first step to formulate a problem statement is to identify what your problem is. Then, decide what you would like the outcome of your problem to be. Lastly, combine the problem with your desired outcome to form your statement.
Problem statement is clear, specific & feasible statement that the problem exists. It has evidences of variables that led to problem. Research Question are those that the researcher would like to answered or addressed in the study.
If you are writing a formal essay, yes. However, in a formal essay, the problem will be the thesis statement. The problem or thesis statement (Note: So there is no confusion, the problem is not the same as the thesis statement.) should be at the end of the introduction paragraph