Does having a PhD give you much advantage for officer promotions?

Unless it's specifically needed for your job, I don't think it gives anyone an advantage. Unless that degree requirement is built into your career path (e.g. a professor, researcher, etc.), it really adds no value to what the Army needs from its officer corps. I earned my PhD through long-term civilian schooling, so it was needed in my career path (professor). I never heard of any officer in a more tradition career path who had a PhD that actually helped them get promoted.

Not really. MIL ED is weighted more than CIV ED. It can’t hurt but it doesn’t necessarily mean you’ll make BZ either.

I have a Bachelor of Science in Business Administration, Masters of Business Administration, Juris Doctor of Law, not one of these was service paid. I was non-scholarship Army ROTC, but still a DMG, and spent most of my career in the Army Reserve for a total of 31 years. The civilian education was important to meet the minimum requirement for education. Your military education path is the most important component for your career. The military will provide / pay for your Masters as part of Intermediate Level Education tied to Fort Leavenworth, KS. If the military wants you to obtain a doctorate they will likely pay for it as part of program that is beneficial to you and the service.