Touching on the second question, since the ship would never actually reach the speed of light, the trip would not seem instantaneous to the people on board. However, the trip would seem much shorter to the people on board than it would to external observers. The people on board the ship would experience length contraction in the direction of travel making their destination closer to themselves, while external observers would notice the people onboard the ship moving slowly, ie, experiencing time at a reduced rate. Either way, the effect is that the people on board perceive the trip to be much shorter (in terms of both distance and time) than an external observer watching their ship. In principle you can get the perceived length of the trip (both distance and time) to approach but not equal zero, though in practice this would involve killing everyone on board and destroying the ship (and maybe even the galaxy).

I agree with the other commenters that the people on board will experience a consistent acceleration of 9.81 m/s² in your described scenario. It might help, conceptually, to imagine an external observer watching someone on the ship jumping up and down at this near-light speed, taking into account the severe time dilation they'd be experiencing: The difference in perception comes because, from the external observer's point of view, the person on the ship is moving in extreme slow motion.

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.

Anything with mass can't travel at the speed of light, but a massless particle, such as a photon, completes its trip instantly from its perspective. A photon is created, departs, and arrives at its destination simultaneously.

What I remember from watching discovery years ago is simply that "time" will slow down so that it takes longer and longer for that object to reach the speed of light. This is the only kind of " time travel" that is theoretically possible.

2. Not really.

Take for example a star 1 light year from sun. If we start this theoretical machine, it WILL take 1 year for that machine to reach there. Same for 100 light years distance. That's the amount of time it takes for light to travel that distance.