Did requesting more info/contacting admissions too early hurt my chances of getting in?

I am 100% sure you did NOT hurt your chances of getting in merely by requesting additional information. Asking for clarification is never a negative. I expect the admissions office would simply send you links to the school's webpages that give the information you requested, and end their email with "Feel free to email us if you have any more questions".

I have been part of the admission process of two two-year long master-level courses in the Management fields. While these are not the same as graduate program in the US (e.g. MSc + PhD program), as far as your question goes, I think there is little difference.

I was specifically involved in responding to emails directed to the admission office and I am confident no damage came to you from asking questions.

The reason being that the person who reads the email and handle this kind of questions is an administrative employee which rarely, if ever, speaks with members of the admission committee directly. Those people are the ones who will make the decision w.r.t. your admission, they are typically busy professor and they never got remotely close to reading an applicant emails, as any menial task is kept away from them.

Emails first pass through the employee, who reads them and replies to all standard questions or questions he has been trained to answer (like 98% of the times).

If answering requires a higher level of responsability/expertise, said employee will ask the course admin/manager (who is typically not a professor, but that might depend on country/institution), yet this is done usually without any reference to the name of the specific applicant (because the information is irrelevant to answer the query).

Furthermore, the reason admission offices put their email out is so that people can ask questions, any questions, so to ensure that the application process is as easy and smooth as possible on both side.

I'd go further to suggest that the more information you have, the better, because part of your application will be to convince them that you'll be an asset to their program as an apprentice of sorts and then possibly later on to partner on research, projects, and writing while you're a grad student. It's essential to make sure your interests and ultimately your personalities match. I'd go so far as to schedule a visit or talk to the faculty you'll be studying under, if you've done your background work.

This is an exciting part of your life, and graduate studies are less about getting a job and more about finding a compatible place to grow and work.