What affiliation to put on an academic paper for alumni authors?

The standard practice is to list the affiliations under which the work was performed. If you performed the work as an undergraduate at your undergraduate institution, then you should continue to list it in work related to that effort. However, you can "update" your address by listing a "current address" along with the old affiliations.

I have published together with authors that only used their personal home address + email address as contact information, so your second option should not be a problem. You can publish in scientific journals without a formal affiliation.

However, if the work was performed at a previous location (e.g. as a student) where you are not currently working, you should include both the previous affiliation along with the current address (as others have also suggested).