Could you post the EXACT terms you are looking at. Because I think you may have mis-read it.
Every credit card company I am aware of has a Minimum Interest charge, depending on the company that can be anywhere from $0.50 to about $2.00. But that only gets applied if you have a balance from the previous month and the amount of interest you would be assessed is lower than the mimimum. For example, if the previous month you had a balance and would have been assessed $0.79 in interest, they will instead charge you the minimum of $1.50.
Now as for the late payment. They do tell you when your payment is due. When you get your on-line statement(or a paper statement) there is a DUE DATE listed. No company is going to constantly remind you. Some have "payment reminders" that you set up, but that is something YOU must take the inititive to do. Depending on the bank you may be able to set up notification to your e-mail, text messages, or even having them call a phone number.