You’d feel nothing out of the ordinary whatsoever. The starscape outside the ship would look strange though, shrinking into a small, blueshifted patch of sky straight ahead, while stars behind you would redshift out of the visible range. Everything moving at very low speeds relative to you would indeed appear to happen really fast.

What are the chances of hitting a small meteorite or part of it, traveling now at relativistic speeds wrt you?

There's about one particle of dust per million cubic metres. c is about 300 million metres/second. So even at 0.5c that's still a lot of particle collisions per second, each having significant kinetic energy.

Basically it would be like flying through explosive sandpaper. Each dust particle would be reduced to plasma, which creates problems of its own.

If you're accelerating there's also the Unruh Effect, which will raise the perceived temperature. By a lot.

There's no way to make this work with any kind of engineering we know about today.

The Unruh effect is theoretical, and no evidence at all has ever been found that it's real. It literally exists as nothing more than a hypothetical mathematical model, that also happens to be debated by others who know enough to effectively debate it, and disagree.

Micrometer-scale specks of dust would hit you like they were armor piercing tank gun rounds. The usual shielding proposed is ice. Lots of ice.

Why ice?

Mostly that it's plentiful, ablative, expendable, plus good radiation shielding (yeah, cosmic ray protons are really going to mess things up at relativistic speeds too unless there's enough mass to stop them).

Extremely low. Space is very empty.

It's one of those cases where you have very small numbers multiplied by very large ones. The actual risk is hard to intuit because there are so many orders of magnitude involved in both directions.

In any case it's probably a moot concern as long as we are living under the twin tyrannies of Newtons Third Law and the Rocket Equation. Building a rocket that can accelerate constantly and noticeably for weeks, months, or even years on end in order to accelerate up to a velocity where Relativity starts to matter requires an absurdly large rocket. Like converting the mass of Jupiter into rocket fuel to make it to the next habitable solar system in a couple of centuries level of craziness.

But it's also very big, and GP doesn't even specify how far of a trip they're asking about nor how small a meteorite.

"Extremely" and "very" don't cut it here. This is beyond the human ability to guess. You'd actually do at least some back-of-the-napkin math to give a real answer, and with a far enough trip, the answer may well become "Almost 100%".

And at a high enough speed, the impacts from the ~1 hydrogen atom per cm^3 in interstellar space become a major problem.

How far a trip: maybe start with the nearest star.

Micrometer-scale dust particles would in fact hit you all the time. And they’d absolutely mess up your ship over time without a lot of shielding.

zero if you hit that spice first

If the light behind you redshifts out of the visible spectrum, would the light in front of you blueshift into dangerous territory? X-rays, gamma rays, etc?

Yes, and this provides a nice intuition about the relation of wavelength to energy. But x and γ wavelengths are several oom shorter than visible light, so you'd have to be traveling at very close to c to experience that amount of Doppler shift.

Yes. To some degree.