On my other boat there is a metal tube plumbed in to the bottom of the pan with an oil change pump attached.
Hot but largely because the oil flows better. I just did mine yesterday and despite warming the engine up it still took 40 minutes for the oil to drain out. Note that me engine has a drain hose plumbed into the pan and oil removal is done through the 1/8" ID hose by gravity flow, so it is SLOW. Cold the oil would take hours to drain through that hose. On my other boat there is a metal tube plumbed in to the bottom of the pan with an oil change pump attached. Warm that engine can be drained of oil in under a minute.
Neither. I have always been told (although via auto techs many years ago) to run the engine to warm up the oil, let is sit for a little while, then change it warm.
But hey... what do I know. I assume different manufacturers probably have different ways the recommend. May want to check your service manual or call a diesel shop and see what they say.
Hot. But i always toss down another quart as it finishes draining out as a 'final flush'. Purely personal opinion as to its effectiveness.