Can someone be ordered to buy a non-required item using personal finances or face punishment, as in the case of a USAFA graduation ring?

While I have not been in the military for many years, back in the day, you received extra pay to cover misc items. Realize items where out, get lost and in some situations are required for the role you are assigned. Unless the items is beyond your ability to obtain I would get it, if you can not afford it, go to your leadership there is usually funds available to help in special cases.

I'm a pv2 single and I have to buy all kinds us extra stuff buy my meals from stores my advise is buy what they tell you to its not worth pissing off your leadership

While it is expected and normally required to purchase a ring, many choose to go with the cheapest option available (~$250). There have been several cases though where cadets have made their case to the commandant about what they would rather do with their money than buy a ring. Some of those cadets were granted an exception, some were not.

You'll probably run into things like this several times in your career, although probably not with items as potentially expensive as the class ring. Many squadrons will ask you for "landing fees" when you arrive in order to cover the cost of your patches, squadron shirt, nametags, squadron upkeep and other similar things.

No.

They cannot, unless in specific cases like MAJ Carl Ballinger has stated below. If you are supplied a stipend for certain expenses, you are expected to spend that stipend on certain things. As far as the general military goes, no, you cannot be ordered to purchase certain items (i.e a command T-shirt, ball cap, ring, belt buckle). Such things would need to be supplied to you.

Definitely not. It's like ball tickets, you're not forced to buy a ticket.

A shipmate of mine was "ordered" to purchase a cell phone. He never had one and didn't want one (one of those off the grid types). He got a couple counseling chits and eventually bought one of the most simplistic phones available. I think it used "units", like 1/10th a unit per message.