A Scenic Trip as a Spaceship at 0.999...c, and questions that arise

[Halc]

Aren't objects far away still moving away faster even with a constant expansion of space-time?

Yes. The rubber band thing illustrates that pretty well. The post by jbriggs444 just above is spot on. Constant expansion is like the end of the rubber band moving away at constant speed.

The current rate of expansion is something near 70 km/sec/mpc. Constant expansion does not mean that the rate expressed that way is constant. It means that something a megaparsec away moving away at 70 km/sec will continue to move away at 70 km/sec forever, even though its 2 mpc away in 14 billion more years. Accelerating expansion means that some distant galaxy will gain recession speed from us.

So the expansion rate might be 70 km/sec/mpc now, and will eventually go down to something like 57 km/sec/mpc and become a constant of sorts, which will indicate exponential expansion, not linear expansion.

If you take points at twice that distance, and per one unit of that distance the expansion rate is constant, then those two units-distanced points will still move away from each other twice as fast.

As measured in cosomological coordinates (comoving frame, proper distance/speed), yes. In such coordinates, speed is more of a rapidity and adds with normal addition (*), not relativistic velocity addition like you'd use with inertial coordinates. So there's nothing funny about recession rates at arbitrarily high multiples of c.

(*) Not as simple as that, but it works for objects with negligible peculiar velocity like pretty much any galaxy.

 

[A.T.]

Am I confusing acceleration and speed here?

I think you are confusing points of space with objects moving locally relative to them.

Here is the rubber band scenario for non-accelerated expansion, where you can always reach any point, given enough time, no matter how much faster than your local speed it initially recedes from your start point (as mentioned by @jbriggs444):
https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope