Is iOS 12 still supported? Now that iOS 13 is out, and iPhone 5S and iPhone 6 won't be getting it - does this mean that the phones are no longer supported?

Should these devices be considered past their end-of-life? I know that Apple released a iOS 12 security update alongside iOS13, but will that continue to happen?

Is there a way to find out the security status of each iOS/iPhone release?
There's a difference between hardware support (new battery, replacement screen, etc) and software support.

Typically, Apple provides hardware support for 5 years after manufacture on their products. 

The iPhone 5S is not currently listed as 'vintage'. However, it's possible the list may be updated shortly, as the phone was discontinued 5 years ago.
The iPhone 6 was discontinued 2 years ago, so will still receive hardware support until 2022.

As for software support: obviously, the devices will not be offered new OS updates, but in the event of a major security flaw, they may receive a patch to fix the problem.

The phones will of course continue to work as they are. Apps will not be updated if they require newer OS versions than are present.