I'm scheduled to graduate next week but my professor has gone MIA due to a death in the family... what can I do?

My question is, is there anything else I can do to push this professor to respond to me and to set up a time to conclude my research project?

I recommend that you don't send your professor any more communications than you already have, and in this case, it is completely inappropriate to "push" him to do anything. The death of a loved one in one's family is a major traumatic event, and a large degree of sympathy and forbearance is appropriate. With respect, it is a much bigger deal than what are relatively minor concerns over graduation.

At this point you have already sent multiple emails, and it is clear that the professor is not able to perform his work duties right now. Rather than contacting him again, contact the Head of Department, describe your case, and request a plan to have your assessment pushed forward in the absence of your professor (e.g., with another supervisor allocated). Make sure you write your request in a way that is sympathetic and kind to your professor, but which acknowledges the reality that he is not presently in a position to assist you. If you have been advised that there is nothing they can do, then you have been advised wrong, because that is clearly bullshit - a Department Head has the power and responsibility to replace incapacitated staff with other staff members who can do their teaching/supervision work.

If the Department is unable to assess you prior to your expected graduation time, this might end up being a case where your graduation is delayed. Accept that and put it in perspective. No-one you love has died, and the worst possible outcome for you here is that your assessment gets delayed, and you end up graduating one session later than you otherwise would have. If this occurs, and you are worried that it will hamper your ability to apply for later positions (e.g., jobs, post-graduate degree, etc.), ask your Head of Department to write a "to whom it may concern" letter that you can give to potential employers, explaining that you are due to graduate at the next conferral and that your graduation was delayed due to a staffing issue through no fault of your own.

One last thing: remember to buy your professor a nice card or some flowers consoling him for his bereavement.

To answer a specific question:

What happens if he doesn't issue me a grade?

In a US-like university system (which based on your terminology I'm assuming), what would normally happen is that you'd be temporarily assigned a placeholder grade, possibly I (for Incomplete) or possibly something else that indicates some sort of administrative delay. Your degree wouldn't be officially conferred, but in all likelihood you'd still be allowed to participate in the graduation ceremony. When the professor eventually returns to duty, you can complete your presentation (remotely if necessary), and he can assign you a grade. That grade then replaces the I, and from then on it's as if you had that grade all along. Assuming it's a passing grade, you will then be awarded a degree, possibly backdated to the end of this semester.

This is a similar process to what would happen if you became ill or incapacitated and were unable to take your last exams on time.

In particular, this sort of situation does not result in you having to retake the course. I would not worry about the possibility of having to attend the university next semester.

If the professor is out for a really long time, or permanently, (which I wouldn't expect in this case), the department can typically have some other professor review your work and assign their best estimate of a grade.