This is a fun question, because it reveals something about English that I never noticed, even after speaking it for 42 years or so.
Your professor is partially correct, but there's more to it. "At your age" is a set phrase with a negative connotation that almost always means, "...you should/shouldn't be doing the thing I'm about to mention or just mentioned." Another common construction using "at" is "at the age of X," e.g.:
He left home at the age of 16.
("At the tender age of...", indicating youthful innocence, is a common cliche.) "At the age of" feels like formal or written English, whereas "at your age" is casual speech (and maybe a little dated). You can also omit "the age of" and just say, "He left home at 16," wherever it would be unambiguous to do so.
Now, for your first sentence:
When I was a student ______ your age, I tried out for our school volleyball team.
Neither "of" nor "at" sounds remotely natural to me. The best words to put in the blank would be nothing at all ("When I was a student your age..." is perfectly natural English) or "about."
"Boys of your age" and "boys your age" are both acceptable. However, "When I was of your age" is not. I'm having trouble figuring out what the rule is, though. Anyone?
Finally, I've never heard "sweetheart" used as a verb, but it sounds so nice I wonder if you're quoting a famous line.