I'm seeking a way to make payments without placing my name and mailing address on file. For example when donating money. In person, cash would work. Online / distant institutions often encourage credit card payment, but that includes name and address. If I order a cashier's check, will it have my name on it?
There are two types of anonymous donations in the United States:
For example if you want to donate $50,000 to upgrade the science lab at the high school you graduated from 20 years ago.
The charity knows who you are but you don't want your name released to the public. The annual guide that the charity publishes lists your donation as being from anonymous. You give them a check/credit card and instruct them to list you as an anonymous donor. They will generally list your donation but they may also characterize it as being from a former student.
The charity has no way of knowing who you are. Cash works, but you have to make sure it makes it there safely. The charity may have additional paperwork to document the large amount of cash. A check could work, but some name would appear on the check. A cashiers check would be somewhat anonymous except to make sure there aren't problems with a cashiers check the general guideline is that the check is made out to "charity or account owner", so that if there is a problem the account owner can redeposit the money.
If option two is the goal, then if the donation isn't done with cash you will still have to instruct them to list it as anonymous. Otherwise they might list the name of the fake corporation which could expose you.
If is completely anonymous, for example a gold coin placed in a donation kettle, you might not be able to deduct it from your taxes. The charity has to give you a receipt for donations over $250, which of course you wouldn't have.