Tried my own last night, apparently the dome light(s) don't go out until you turn the key, just inserting it doesn't kill them. So in your case the lights would go out after you close the door and the delay times out, and then come back on when you turn the key on.
I don't have my EVTM available but as I recall the circuit; the (a)relay mentioned previously, the (b)door jamb switches, the (c)key-in switch/relay, and the (d)timer module all have a part to play for lights to come and go off. Your test show that b, c and at least the delay part of d are ok. The timer d also works the intermittent wipers and the windshield washer pulsing, so you could check those features (although there are several more) easily.
A cursory check of relay coil can be accomplished just by putting your finger on the relay, and either operating the ignition from off to run (but not start), or operating the door jamb switch by hand (and delay expires), to feel if the coil energizes.
