Even though we can perceive the "depth" of the stars through motion in this simulation, every star is still going to look equally far away, because their positions won't change between the left and right eyes. The same way we can't perceive the sun as further away than the moon -- depth perception merely tells us they're both "max far away".
You could always introduce depth perception by faking your eyes to be light-years apart, but of course that's not how it would really be in the spaceship...
https://en.m.wikipedia.org/wiki/A_Slower_Speed_of_Light
I also found this render (and game!) that show off some aspects of relativistic spaceflight. https://youtu.be/Ix1XlxF66Zk?si=8HzDH3U-CRM-erRB
I think the technical limit might be moot, better question might be how strong of a stomach you have, geometry gets pretty wacky near c
You won’t feel the acceleration ;) nor the light intensity of the higher velocities (luckily).
Years are passing in seconds in this websites simulation, if you were experiencing it all in "real time" everything would appear pretty much stationary for quite a while.
Even at like 95% the speed of light, you're talking about years of time required to move between stars - nothing will appear to change much day to day until you actually get close to one of them.