How many papers should a Ph.D. student review for their professor per year?
There is no hard and fast rule here: some professors don't ask their students to help review at all, others pass all of their reviewing to their students.  I think the latter position is generally indefensible---the professor is being asked to review because of their expertise, not their ability to press-gang students into doing work for them.

In my own thinking, the driving principle should be the education of the student.  Every person who is aiming for a career in academia will be asked to review, and every person who authors papers should have a chance to experience the reviewing process from the other side as well.  To me, though, that calls for a few papers a year, not a paper every week.

From your description, I'm not quite sure how to judge the situation with your advisor.  At first reading, my thought is that your advisor is accepting more reviews than they can handle, and dealing with the situation by passing on the reviews to you and your fellow graduate students, which would be inappropriate.  If your advisor is actually spending a lot of time with the paper as well, and essentially co-reviewing with you, then perhaps they just have a very unusual perspective on how to learn about paper-writing and the state of the field.  

I think it also depends a lot on the quality of papers you are getting asked to review: if you are being asked to read good papers sent to top journals, then it's more defensible, since you really are seeing the newest relevant literature before anybody else; if there's a lot of low-quality stuff to iffy venues, though, it's probably a waste of time for everybody involved.

Bottom line: you're being asked to do an unusually high amount of reviewing, and if it's interfering with your ability to pursue the rest of your program, then it would be reasonable to bring it up with your advisor and ask for a change.