STN, LCY or LTN from Gatwick

Easiest is actually Luton. This is on the opposite side of London, but paradoxically you can get a direct train from Gatwick to Luton on the only rail line that goes through London, "Thameslink": Brighton - Gatwick - London - Luton - Bedford (with a few other stops). For Luton airport you get off at Luton Airport Parkway station and there is a shuttle bus to the airport.

But, now the bad news. If you land at Gatwick at 23:45, it will take up to an hour to exit, get through immigration and get out of the airport (00:45). You probably (depending on airline), want to be at Luton check in desk at least 1 hour before departure, better 90 mins. or 2 hours = 05:00. That is both too late for the last train out of Gatwick, and too early for the first train in the morning.
Where are you going to sleep?
The only way you can do this is with a pre-booked taxi, which will cost mega £££ at that time, and take ~ 1½ to 2 hours to any of these airports. Arrive at STN, LCY or LTN 00:45 + 90 mins. = 02:15, have to be at check-in ~05:00 CONCLUSION - no sleep.

There is no way I would do this!

That is both too late for the last train out of Gatwick, and too early for the first train in the morning.

Where are you seeing this information? When I check the schedules at nationalrail.co.uk, it's turning up hourly trains throughout the night. For example, there's a journey leaving Gatwick at 1:21 and arriving at Luton at 3:31.

I'm just intrigued what destination has a 6:30 departure from STN, LCY and LTN but not LGW.

I stand correct, I believe you if you say there is a 24 hour train service. But that still involves travelling to Luton airport arriving after 03:30 for a 05:00 check-in. When is aklittle7 going to get any sleep?

"When is aklittle7 going to get any sleep?"

Maybe on the plane and/or the train and/or the airport. Or maybe they're coming from a timezone far enough away from GMT that they're not going to be expecting to sleep during the UK overnight hours. Or maybe they're one of those people for whom the occasional night without sleep isn't too painful. Or maybe they know it will be painful but find the plan appealing for other reasons.

Haha! Yes it's one of those circumstances where we have no choice other than to deal with a miserable night of bad sleep. We are arriving from Norway to london, then we have to fly to Dublin to catch our flight to the US. It's going to be a rough night, but we have no other option.

So Luton is the best bet? And where do we get train tickets?

so this mad plan is to go Norway > Gatwick > some other airport in London without sleep > Dublin > home?

Why don't you fly from Norway to Dublin and cut out the craziness in London? Or fly Ryanair from Gatwick (08:50 - 10:10) straight to Dublin.

get train tickets at train station at Gatwick.

....and buy the ticket with destination Luton airport so the bus fare is included from the rail station to the airport.

I'm sure it's too late, but Icelandair offers great cheap connections to much of the USA from Oslo, about 9-12 hours, and then Norwegian offers cheap nonstops to a lot of places Icelandair doesn't go, like Vegas, LAX, Bay area. What you are doing is pretty dumb but if you are young it doesn't matter much.

Our flight to the US leaves Dublin at 12:00 and I was told that we needed 3 hours from when we land in Dublin to go through customs/immigration etc. Otherwise we would not be doing this insane schedule. We would've just taken the flight from Norway that gets in to Dublin at 10:50 if we could.

Be aware if on Norwegian into Gatwick that it is in the bottom 3 for punctuality record of any scheduled airline operating at a UK airport (the only ones worse in this regard are Air Transat and Icelandair).