Eight seconds after powering on, my phone goes dark and becomes unresponsive. What can I do?
I have owned several Samsung galaxy devices (including an S3 which is somewhat similar to your phone). Often I find that the battery becomes faulty after 2 years of usage. The symptoms you mention also sounds like battery related problems. The key giveaway is that you unable to sustain download mode before your screen goes dark. 

Troubleshooting


Try replacing the battery. This will eliminate the possibility of a
faulty battery. After replacing the battery, see whether your phone
will stay on for more than a few seconds.
If you unable to acquire a second battery,  try booting your phone whilst in the charger. It should be noted that using the phone
whilst charging isn't usually a good idea, however for
troubleshooting it should be fine. 
Check your charger, wires should not be frayed or worn. Worn
chargers can affect your phone's battery.