What I have learned from conversations with people who use less common pronouns (typically they/them) is that there is no expectation for perfection or magical divination of pronouns. There's no penalty for making a mistake, only a hope that you'll do your best when made aware or corrected and not argue or dispute.
Occasionally, you may hear that it is not acceptable to make any mistakes, that you must avoid being tricked into using the wrong pronoun and revealing yourself to be bigoted and deserving of permanent shame and punishment. In my experience, this message does not come from people who are actually interested in respecting people's pronouns. Rather, it comes from people who want you to believe that describing people as they want to be described puts an impossible burden on other people, or who want you to think that a trans or nonbinary person is a scary threat.
I think it's appropriate due diligence to notice on someone's academic profile online whether they specify particular pronouns, and if so, use those whenever you use a pronoun to describe them. If they aren't stated, though, you don't need to investigate any further to make sure you get it right, just try to use the best information you have at the time.
That said, you can never go wrong with "Dr. Doe's work". "Their work" is also fine both as a way of describing an individual whose pronouns you don't know and also in a plural sense that it's rare (at least in my field) for academic work to be a solo project; "their work" can acknowledge that papers published by Dr. Doe are a group effort and when you describe work by Dr. Doe you really refer to work by Dr. Doe and their lab/collaborators. You could even do this explicitly by referring to work in Dr. Doe's lab (if appropriate to your field) rather than only by Dr. Doe.
I would not recommend using other newly created pronouns (Xe/xem, Ver/vir) unless someone specifically requests them to be used; these are specific pronouns and unlike singular "they" do not have the same role in describing someone whose gender you are unaware of.
I think if you are referring to the same professor by name more than once or twice in your statement of purpose you are probably over-referring to that one individual, regardless of how you refer to them.
Some old guides recommend using the masculine gender in English as a generic gender or flipping between masculine and feminine; I would definitely avoid assuming masculine gender for everyone because of the stubborn implication that professors must be men. I would also avoid simply alternating genders when you're talking about actual people; it's appropriate for example/hypothetical cases where gender should be unimportant (e.g., labeling characters 'Alice' and 'Bob' in describing an imaginary interpersonal interaction), but is not a suitable strategy for a specific person whose gender is unknown to the writer/speaker: it's better to use singular 'they' in that circumstance.