TerrorBite

You can think about it this way. In relativity:

• You’re not allowed to have any way to determine an absolute speed. If your perceived acceleration were to vary (for a constant thrust) depending on your speed, that would give you a mechanism to determine absolute speed, but absolute speed doesn’t exist in relativity.
• Rather than “nothing can go faster than the speed of light,” given that we’ve just determined that absolute speed doesn’t exist, the next rule is instead: you are not allowed to observe anything travelling faster than the speed of light relative to you, and relativistic effects will ensure that this is so.

My main question, and the one that I initially came here to ask, is: if their ship continues applying the force that, under classical mechanics, was enough to accelerate them at 9.81 ms^-2, would the people inside still experience Earth-like artificial gravity, even though their velocity as measured by an observer is now increasing at less than that rate?

Relativity says yes. There’s no absolute speed, only relative speed; within the local reference frame of the ship, everything will continue to work normally, including the force experienced due to acceleration.

My understanding is that a trip taken at the speed of light would actually feel instantaneous to the traveller, while taking distance/speed of light to a stationary observer.

The ship is not actually going to reach the speed of light (as seen by an outside observer) though. The faster the ship goes, the more its (observed) mass increases, and the 9.8m/s² acceleration will have less and less of an effect. But to the people inside the ship, it appears as though they can accelerate indefinitely, going faster and faster at their steady rate of acceleration. Due to relativistic effects, it’ll never look like they are passing any objects outside the ship at more than the speed of light; instead it will appear as though the distance they have to travel is compressed, so they don’t have to travel as far.