Many of the issues you described stem from forgetting one key fact: Velocity is not an absolute. It is relative.
Ships flying at 1000 km/s are not inherently immune to projectiles flying 1 km/s if those proectiles are fired by another ship going 1000 km/s in the same direction, because the projectiles' 1 km/s is added to the firing ship's 1000 km/s. If it wasn't, then firing a forward-aimed gun while in motion would be suicide - you'd hit yourself because you're moving faster than the projectile you fired.
To put it another way, a bullet fired from an AK-47 has a muzzle velocity of 710 m/s. I am currently moving at a velocity of approximately 30 km/s (relative to the sun), or about 42 times the bullet's speed. Does this make me immune to gunfire? No, because the gun is moving at the same 30 km/s (relative to the sun) as I am, and the 710 m/s of the bullet's muzzle velocity is added to that, not an independent quantity.
To have a prolonged engagement (longer than a single salvo as they fly past each other), ships don't need to "slow down", they need to match velocities - but the final matched velocity can still be absurdly high (relative to the local star, or planet, or whatever other reference point you might prefer) as long as they are more-or-less at rest relative to each other.
Regarding evasion, you're right that it's extremely effective against dumb projectiles at long ranges, but this is purely a function of the time it takes for the projectile to reach the target and the rate at which the target can accelerate to evade. 898
The velocity of the target is not a factor, aside from how it affects the projectile's time to target - a very high velocity towards the attacker makes evasion harder, not easier, because it reduces the available time in which to evade the attack.
And that pirate who's running dark? They can still be moving at a very high speed because, in space, you only need to burn engines to change your velocity, not to maintain it. So, if they know that ships tend to pass through an area with a specific velocity (both speed and direction are relevant here!), they can match that vector at a distance beyond sensor range, then coast through the target area with engines off until they notice potential prey nearby and fire up the engines to make minor course corrections and approach.
In overall conclusion, though, deep-space combat, as a rule, isn't entirely realistic in the first place, unless the aggressor already knows where exactly their target is going to be, either because the target is following a known trade route or because they have intel on the target's planned movements. Why? Because "Space is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space." (<PERSON>)
The Earth's oceans are microscopically tiny compared to even the volume of space within the moon's orbit, never mind the solar system as a whole or looking at interstellar space, but, even so, terrestrial naval battles tend to happen near ports or other significant locations, or along major trade routes for things like commerce raiding, not at random locations on the high seas. Similarly, space combats are most likely to happen near planets, Lagrange points, or other significant locations, not because closer ranges and lower relative velocities make targeting easier, but because it's nigh-impossible to even find your target anywhere else. 947