Should I try to include $5.67 interest on checking when I didn't get a 1099-INT, or I assume it's too piddly to include. Too piddly?

You have to report all your income, no matter how "piddly" it is. Enter it as if you did get a 1099-INT. All you have to enter is the name of the bank and the amount.

The bank is only required to report amounts over $10.  You could include it without the 1099INT.  It isn't going to change your tax in any meaningful way.