Does it turns on immediately and stays on until break plus the delay time requested?

Hello.This relay has a post purge manually adjustable delay from 12 to 390 second.Let us know if you have any other questions.Customer Support Team of PartzStop.

If trigger power is restored during the delay, will it still cut off when the delay time is reached?

The question is a bit ambiguous with out a wiring diagram. The normal application is to extend the on signal. This is a control relay, it does not power the fan directly. The unit extends power on "control" by up to 5 minutes. If the trigger is off due to power outage the relay is off too. The timer starts over when power on is restored and the trigger is high.

would this work on dc?

I am not sure if it work with DC. There could be internal circuits that could get damaged with direct current. You may want to submit your question in their website. I believe it is www dot icmcontrols dot com

From the 12 second setting how many seconds are added with each adjustment

Considering the graduation, does not look like it is linear! If I have to guess, it maybe 40 seconds each marking after the 12. After number 12, there are four more divisions then number 200 then again another four and last number is 390. Go figure!!