What would be the maximum trip time on ground (train or ship) which is competitive with flying + a hotel night at destination?

I'm going to guess that you aren't in the US because you didn't mention driving. This question has been debated several times here on FT. Personally, four hours is my limit, all in. If I can get to the airport, fly, and get to my destination faster than four hours will drive me, then I'll fly.

For business, it's around 3 hours for me, though my business rarely takes me anywhere within 8 hours of our office.

For pleasure it's a bit more difficult balance. Our most frequented leisure travel location is Orlando and we fly whenever there are very inexpensive tickets available, but drive more often, probably 75% of the time. It's around 8 hours from west Atlanta. The 25% that we fly usually lines up with an on-property stay at Disney which comes with a free airport shuttle. We've only once flown and rented a car, before the shuttle was free and we were spliting a condo with friends.

Savanah/Hilton Head is another that depends on flight prices/sales but ends up being around 50/50 since airtran offers alot of $80-$100 r/t and Delta often matches.

My home town in KY is about a 5 hour drive from ATL, but would be around 4 hours to drive to the airport, get through security, fly to LEX, then drive to my parent's house... I've never once flown.

Because you cannot sleep in a bedroom while driving.
A train or a ship is driven. So the passengers can settle down between sheets.

What is the minimum trip time where you would occupy a cabin/compartment rather than sit in public areas?

Still a little confused. You mean how long I would travel in a seat before I would upgrade to a cabin, irregardless of the mode, or how long I would travel in a plane seat versus a train seat?

In the US, I can't imagine ever taking the train outside of Amtrak's Acela between Boston and NY. And there I would say I would prefer the train due to comfort, but if I am going only to NYC will choose the plane either because it is half the price, OR the hassles of getting to the train station outweigh getting to the airport.

Time of travel might be pretty relevant too... I'd be prepared to travel 8 hours in a sleeping compartment on an overnight train - but if it's a daytime trip, I'd much rather fly.

It really depends on the circumstances.

For a night train, it's often a better option, as, if I can get one at midnight, it basically buys me back the evening at home.

For a day train, if I can work from the train, then 4-5 hours is fine (particularly on the ones that have Wi-Fi connectivity). Beyond that and it depends on what I'm doing around that time.

It'd be great to have more sleeper trains running from London, particularly through the Channel Tunnel - so much nicer than the old 4am start.

I wouldn't consider driving an option, really - I don't drive where I don't have to. Plus, it's lost time, really - 4 hours on the road listening to music vs 4 hours on my laptop on the train in peace and quiet...

Considering I would have to drive at least 1.5 hours or take a bus just to GET to a train station, I would be much more likely to fly at that point (as that is the same distance to the closest major airport, JAX). My time is precious, and I don't like wasting it when I could be enjoying it. While it is relaxing to me to be on a train, I can sit and read at home - I would much rather be visiting places. A drive is a little different - I can get out and explore if I have the time.

In which case, you would need to eat breakfast and dinner at destination.

How long would you spend at your destination for a hotel night - how many hours before duty time would you want to arrive?