Am I legally required to carry O2? (Full disclaimer: I already carry two completely separate emergency O2 kits for redundancy in emergencies, even if I am not diving or taking divers out.)
No, you aren't legally required to carry O2. In fact, unless you have an 02 Provider cert, you may have some difficulty getting 100% O2 fills. Oxygen is classified as a drug.
Am I legally required to administer first aid in the event of a minor injury? What if my friend refused?
No. Not as a lay person. A professionally trained Dr. or first responder may have a duty to help, but even then, the victim may refuse aid. Some of this is covered in a First Aid/EFR course. Which I would recommend taking anyway. One of the things that was made pretty clear in this course is that you need to get permission from the victim to help. One caveat is if the victim is unconscious. In that case, consent is presumed.
There are some that have divers on their private boat sign a waiver. That probably doesn't hurt too much, but not ironclad either.
You are the captain of the boat. A captain has a number of obligations and responsibilities, which are the same also if passengers are not paying.
Did not you learn about captain's responsibilities when you did follow the course for getting your captain's license?
A Captain's license is required for a for-hire vessel. A Captain's license is generally not required for a private vessel. There are rules around contribution amounts to keep the trip private vs. charter. Authorities in some areas have been cracking down on illegal charters. A rule of thumb is that the passengers can contribute to a share of the expenses. The owner can post on social media to take out additional divers with an expectation of a share of fuel, ice, etc. costs. Posting a set amount as a fee is where it gets into trouble.
And as far as courses go for a private vessel. It depends. I technically do not need to have taken any course or need any license to operate my boat. I did take a course, but it isn't required for me because I've been grandfathered in. Law was passed in FL requiring education, but everyone born before that date is exempt. Even then, it's not a license. It's an education requirement.
Also your authority is truly absolute: everyone on board must simply follow your orders. No exception...
Agreed with this 100%.